I. Plantinga's Reformed Epistemology and His Mature Account of Warranted Belief
Since the 60s, Alvin Plantinga has been arguing that belief in God is "properly basic". That is, like belief in material objects, the past, and other minds, belief in God can be rational in a direct, non-inferential way, wholly apart from propositional evidence and argument. This thesis constitutes the core idea of his version of so-called "Reformed Epistemology".
Plantinga's mature defense of his thesis is grounded in a proper functionalist version of epistemic externalism. Plantinga summarizes his account as follows:
"Put in a nutshell, then, a belief has warrant for a person S only if that belief is produced by cognitive faculties [e.g., perception, memory, introspection, reason, and testimony -EA] functioning properly (subject to no dysfunction) in a cognitive environment that is appropriate for S's kind of cognitive faculties, according to a design plan that is successfully aimed at truth."
So that's what's required for a belief to have any warrant at all on Plantinga's account. But he allows that warrant admits of degrees, and he ties the degree of warrant a belief enjoys to the degree of firmness with which it is believed: "We must add, furthermore, that when a belief meets these conditions and does enjoy warrant, the degree of warrant it enjoys depends on the strength of the belief, the firmness with which S holds it." Thus, for such a belief to have a degree of warrant sufficient for knowledge, it must be held with a very high degree of firmness.
Putting these points together, Plantinga's account can be summed up as follows:
I. Conditions of warrant are not met = no warrant (whether the belief is held firmly or not)
II. Conditions of warrant are met + low degree of firmness = low degree of warrant.
III. Conditions of warrant are met + high degree of firmness = high degree of warrant.
So that's Plantinga's account of warranted belief in a nutshell. But how does this account connect to his account of warranted theistic belief in particular?
II. Plantinga's Mature Account of Warranted Theistic Belief: The A/C Model
Plantinga argues that it's epistemically possible (consistent with what we know or reasonably believe) that God has designed us in such a way that we are naturally endowed with a cognitive faculty -- what he (following John Calvin) calls the sensus divinitatis -- that, when functioning properly in an epistemically congenial environment, spontaneously and reliably produces true beliefs about God. So, for example, when one looks at the starry heavens, the sensus divinitatis is (when functioning properly) naturally disposed to spontaneously trigger the belief, "God made all this"; when doing something wrong, it's disposed to trigger the belief, "God disapproves of what I've done"; etc. Therefore, if such belief meets all of the conditions of warrant -- viz., (a) it's produced by a properly functioning cognitive faculty (viz., the sensus divinitatis), (b) the faculty is successfully aimed at truth, and (c) the environment in which such beliefs are formed is epistemically congenial --, Plantinga's account entails that such belief enjoys at least some measure of warrant. And if (d) such belief is held with a very high degree of firmness, the degree of warrant it enjoys is sufficient to constitute knowledge (assuming the belief isn't subject to undefeated defeaters).
We've now looked at Plantinga's account of warranted belief in general and his account of warranted theistic belief in particular. It is now time to take a look at his account of warranted Christian belief.
III. Plantinga's Mature Account of Warranted Christian Belief: The Extended A/C Model
Very roughly, on Plantinga’s model of warranted Christian belief, the Holy Spirit acts on the believer by repairing the sensus divinitatis from the ravages of sin, so that it naturally, spontaneously, and reliably produces true belief about God in the basic (i.e., non-inferential) way. It also repairs the person's affective equipment, so that it is no longer hostile to God and his purposes, but is rather attracted to them and delights in them. Finally, the Holy Spirit functions as an analogue to a properly functioning cognitive faculty by acting directly on the "heart" of a person to produce belief in the core truths of Christianity (what Plantinga calls the Great Things of the Gospel) when they are presented to them (if the person wills to accept the gospel message). Therefore, as with warranted theistic belief as described in Plantinga's A/C model, if Christian belief formed in the way described in his Extended A/C model meets all the conditions of warrant, i.e., (a) it's produced by properly functioning cognitive faculties, (b) the faculties are successfullly aimed at truth, and (c) the environment in which they're formed is epistemically congenial, then it enjoys at least some measure of warrant. And if (d) (due to the internal instigation of the Holy Spirit) the belief is held with a very high degree of firmness, the degree of warrant it enjoys is sufficient to constitute knowledge (again, assuming the belief isn't subject to undefeated defeaters).
We've now seen a sketch of Plantinga's account of warranted belief in general, warranted theistic belief, and warranted Christian belief. What to make of these accounts? I mention eight criticisms below from the literature that have real bite (for more elaboration, click on the links).
IV. Criticisms of Plantinga's Account of Warranted Belief that Have Real Bite
With respect to his accounts of warranted theistic and Christian belief: (i) His analysis of warranted Christian belief can't adequately account for the variability of belief among Christians; (ii) his postulation of a sensus divinitatis in human beings is at odds with the empirical evidence regarding the demographics of theistic belief; and (ironically) (iii) his account entails that the belief of most Christians has little by way of warrant. And of course there's (iv) the Great Pumpkin Objection. But deeper problems lie with his basic account of warrant (see below).
With respect to his account of warranted belief in general: (i) His case for a theistic version of proper functionalism is undercut; indeed, (ii) his theistic version of proper function entails that no beliefs have warrant; (iii) his proper functionalist amendment to straight reliabilism is unmotivated; and (iv) his account of warrant is subject to counterexamples with respect to both to the necessity and sufficiency of the conditions he proposes.
For these reasons, Plantinga's proper functionalism fails to show that Christian or theistic belief can be warranted in the basic or non-inferential way, or even how beliefs can be warranted in general.
V. A Final Note on Plantinga's Methodology
I'll close with a worrisome point about Plantinga's methodology made by Tyler Wunder. Consider the two main types of analyses/accounts of a concept or property Plantinga deploys in his warrant trilogy:
(i) The strict one of logically necessary and sufficient conditions for concept application.(ii) A comparatively lax one according to which the logically necessary and sufficient conditions need only apply to a small set of "core", paradigm cases to which the target concept applies. The "penumbral" cases -- i.e., cases where it's at least somewhat less clear that the target concept applies in the most literal sense -- need not satisfy the stricter standards of satisfying the logically necessary and sufficient conditions spelled out in the relevant analysis.
Plantinga applies the strict type-(i) analysis to views he doesn't like, but applies the relaxed type-(ii) analysis to his own accounts of warrant and of proper function. Pending some principled basis for applying this double-standard, such a method of evaluating his own views vis-a-vis others appears to be illegitimate. Let's see how Plantinga's use of this double-standard plays out with the two key notions at issue here: the concept of warrant and the concept of proper function.
A. Double-standards for his concept of warrant
As alluded to above, Plantinga himself rejects the strict, type-(i) standards for satisfactory philosophical analyses when it comes to his own account of warrant. As he puts it:
"I must acknowledge a complication with respect to my way of thinking of warrant. I aim at something in the neighborhood of an analysis of warrant: an account or exploration of our concept of warrant, a concept nearly all of us have and regularly employ. (As we all know, desperate difficulties best any attempt to say precisely what analysis is.) Thus at the least I should be looking for necessary and sufficient conditions. But I very much doubt that there is any short and elegant list of conditions at once severally necessary and jointly sufficient for warrant. This is a way in which philosophy differs from mathematics; and epistemology differs more from mathematics, along these lines, than, for example, philosophy of logic or the metaphysics of modality. Our concept of warrant is too complex to yield to analysis by way of a couple of austerely elegant clauses."(Warrant and Proper Function, ix)
In the place of the strict, type-(i) standards for adequate analysis, Plantinga states the conditions for the more lax, type-(ii) standards for adequate analysis mentioned above:
"The structure of [warrant], I believe, involves a central picture, a group of central paradigms – clear and unambiguous cases of knowledge – surrounded by a penumbral belt of analogically related concepts, concepts related by different analogies and standing in different degrees of closeness to the aboriginal paradigms. Between the central core area and this penumbral belt there is a more shadowy area of borderline possible cases, cases where it isn’t really clear whether what we have is a case of warrant in the central sense, or a case of one of the analogically extended concepts, or neither of the above; and beyond the penumbral belt we have another area of borderline cases. Hence perhaps a good way to characterize our system of analogically related concepts of warrant is to give first, the conditions necessary and sufficient for the paradigmatic core. (Even here, as we shall see, there is no stylishly sparse set of necessary and sufficient conditions: various qualifications, additions and subtractions are necessary.) Second, what is needed is an exploration of some of the analogical extensions, with an explanation of the analogical bases of the extensions. This way of proceeding is less elegant and pleasing and more messy than the analysis that we learned at our mother’s knee: it is also more realistic. (Warrant and Proper Function, ix)
Thus, as we've pointed out above, Plantinga thinks the necessary and sufficient conditions of a proper analysis need only apply to the “core” or paradigm cases; proposed counter-examples that only apply to the “penumbral” cases are not successful counter-examples to the proposed analysis of a given concept. But Plantinga’s proposed counter-examples to naturalistic analyses of function and proper function appear to involve just the penumbral cases. But if so, then if we apply such relaxed standards to naturalistic analyses of function and proper function, then Plantinga’s proposed counter-examples appear to be unsuccessful.
B. Double-standards for his concept of proper function
Again, Plantinga himself rejects the strict, type-(i) standard of analysis when it comes to his own “intelligent design" account of proper function:
“Perhaps it is true that our concept of proper function doesn’t have non-trivial necessary and sufficient conditions, but if that’s true, it is not in my opinion a defect in the concept; in particular, it isn’t grounds for supposing the concept unintelligible. (Many magnificently intelligible concepts do not have nontrivial necessary and sufficient conditions – for example, red, horse, and belief). (Plantinga, “Reliabilism, Analyses, and Defeaters”, Philosophy and Phenomenological Research 55 (1995), pp. 454-455.)
But if Plantinga accepts such relaxed standards of analysis for his own accounts, then it’s not clear what principled grounds he could offer against the naturalist applying the same standards to naturalistic accounts of proper function. But if not, then since Plantinga's proposed counter-examples to naturalistic accounts of proper function all appear to involve only "penumbral" cases, it looks as though his proposed counter-examples against such accounts are unsuccessful.
 Warranted Christian Belief, p. 156.
 (Plantinga calls his account of warranted theistic belief The A/C Model, inspired as it is by the writings of Aquinas and Calvin.) The following is a rough summary of some key points in ch. 6 of Warranted Christian Belief.
 Question: If all human beings are endowed with a sensus divinitatis, then why do very many people fail to form theistic belief -- at least in the basic, non-inferential way Plantinga describes? Answer: The sensus divinitatis has been damaged by the Fall of Man and human sin. More on this in the next section.
 The following is a very rough summary of some key points in chs. 7-9 of Warranted Christian Belief.
 This part is a bit tricky. For, again, according to the model, the Holy Spirit doesn't produce warranted Christian belief via the cognitive faculty of the sensus divinitatis. Rather, it produces it by acting directly on the "heart" of the person. Therefore, strictly speaking, specifically Christian belief isn't produced by a reliable cognitive faculty, but rather by a reliable process. As you might have guessed, people have raised concerns about this. See, for example, Craig and Moreland's Philosophical Foundations for a Christian Worldview (IVP, 2003), pp. 168-169; Beilby, James. Epistemology as Theology: An Evaluation of Alvin Plantinga's Religious Epistemology (Ashgate, 2005), pp.151-153.
 Cf. Beilby, Epistemology as Theology, pp. 153-156.
 Cf. Maitzen, Stephen. "Divine Hiddenness and the Demographics of Theism" (Religious Studies 42 (2006), pp. 177-191.
 See, for example, Beilby. "Plantinga's Model of Warranted Christian Belief", in Peter-Baker, Deane. Alvin Plantinga (Cambridge University Press, 2007), p. 146; DeRose, Keith."Are Christian Beliefs Properly Basic?" APA Eastern talk, 1998. Available here; Chignell, Andrew. "Epistemology for Saints: Alvin Plantinga's Magnum Opus", Books & Culture (March/April 2002), p. 21.
 Cf. Wunder, Tyler. "Anti-Naturalism and Proper Function”, Religious Studies 44 (2008), pp. 209-224; Bardon, Adrian. “Reliabilism, Proper Function, and Serendipitous Malfunction”, Philosophical Investigations 30:1 (Jan. 2007), pp. 47-64; Graham, Peter. "Intelligent Design and Selective History: Two Sources of Purpose and Plan" (in Jonathan Kvanvig, ed. Oxford Studies in Philosophy of Religion, Volume 3, 2011). (A link to the paper can be found here)
 In addition to my formulation of the objection at the link above, see R. Douglass Geivett and Greg Jesson. "Plantinga's Externalism and the Terminus of Warrant-Based Epistemology", Philosophia Christi 3:2, pp. 329-340.
 Feldman, Richard. “Proper Functionalism”, Nous 27 (1993), pp. 34-50.
 See, for example, Greco, J. 2003. “Virtue and Luck, Epistemic and Otherwise,” Metaphilosophy 34:3, 353-6; Lehrer, Keith. "Proper Function vs. Systematic Coherence", in Kvanvig, Jonathan. Warrant in Contemporary Epistemology: Essays in Honor of Plantinga's Theory of Knowledge (Rowman & Littlefield, 1996), pp. 25-46, esp. pp. 32-33; Feldman, “Proper Functionalism”, pp. 34-50; Senor, Thomas. “A Critical Review of Alvin Plantinga’s Warranted Christian Belief”, International Philosophical Quarterly 42:3, Issue 167 (September 2002), 395-396.