Authorship attributions for any long-lasting text are inevitably layered and complicated. For instance, if you have read the Epic of Gilgamesh, you have read an edited translated reconstruction from several different variations that were themselves in different languages and from different times and cultures, and thus you have read a version that probably no one read before that version was published; editing, translating, and reconstructing all add complications to the question of who authored something, and they also complicate all inferences about authorship.
For all the books of the New Testament, there are 'conservative' and 'liberal' positions about authorship; these are misleading labels for the position that the common and customary attributions are at least approximately right (conservative) and the position that the common and customary attributions are not correct enough to be counted as right (liberal). The reason it has to be put in this form is that common and customary attributions are often themselves approximate and incompletely certain, and authorship attributions even in other literary cases are made with an allowance that, for instance, there could also be intermediaries, editors, co-writers, different revisions, and so forth. For instance, the customary and common attribution for the Gospel of Matthew is that it was written by St. Matthew the Apostle, so the 'conservative' position is that it was written by St. Matthew or close enough; if it turned out (as some Church Fathers thought) that St. Matthew was the writer of the original draft, but that later writers translated it into Greek, perhaps with authorized expansions, this would still suffice for the conservative position. Likewise, traditionally St. Matthew and St. Levi are the same person (people in Judea and Galilee in the first century often had more than one name for various reasons), but if it turned out that they weren't and the Gospel was written by St. Levi, not St. Matthew, this would still be a conservative position, although one that no doubt conservatives would want clear evidence about -- the attribution to St. Matthew is right enough, and any confusion is actually elsewhere (in thinking that St. Matthew and St. Levi were the same, when in reality only the latter was the author). If one held that the authorship of the Gospel of Matthew had nothing to do with St. Matthew, and that the attribution was due not to St. Matthew having something somewhere to do with it but just an attribution that arose later for independent reasons, this would definitely be a liberal position on the authorship of the Gospel. Likewise, if it were a later forgery with no actual causal connection to St. Matthew, even indirectly or loosely, this would also be a liberal position.
Of all the books of the New Testament, the one that perhaps most easily lends itself to the liberal position, and causes the most difficulties for the conservative position, is 2 Peter. (2 Timothy is probably its only rival on this particular point.) There are puzzles with it all the way across the board, however, that are not resolved.
The first and most basic elements to consider are what the work actually says. 2 Peter 1:1 gives as the 'from' of its greetings, Symeon Petros, slave and apostle of Jesus Christ. 'Symeon' is an unusual variant of 'Simon', but it is used of Simon Peter in Acts 15:14; Petros is a special title of Simon Peter in the same sense that Christos is a special title of Jesus Christ; and while the New Testament attributes apostleship to more than the Apostles proper, Symeon Petros...apostolos, is a very distinctive attribution. The epistle refers to a prior letter (without giving any indication as to its content) in 2 Peter 3:1; it refers to "our beloved brother Paul" (3:15) and his letters (3:16), and groups that first letter and some Pauline letters, whichever such letters might be in view (the usual educated guess based on echoes is that at least 1 Thessalonians is being considered), as "writings" (3:16, usually translated as 'Scriptures') that must not be distorted. It also says that "we" have made known Christ's power and appearing not by following stories, but having been "eyewitnesses" of his greatness (1:16), which it links with the Transfiguration, "we" having been brought with him and hearing the voice on the mountain (1:18). It anticipates the author's death as was made clear to the author by Jesus Christ himself (1:14). Thus the self-attribution of the text seems clearly to be Simon Peter; but is (as is often the case in this period) somewhat loose. What it strictly tells us is that it is written and sent under the authority of St. Simon Peter, the Simon Peter, and in the course of the letter it refers, as an eyewitness account, to an event of which Simon Peter was one of very few eyewitnesses. This is consistent with a very wide range of authorial possibilities. For instance, it has been suggested by some that several of St. Paul's letters were basically written by committee, with Paul simply supervising and giving final authorization; this would be equally consistent with what we find here. Many other things are possible.
It's difficult to pin down the earliest quotations and allusions, because any such assessment requires making a number of assumptions, but a plausible candidate is 1 Clement (9:2, 23:3), which would make 2 Peter quite early; if these are not genuine allusions, then the latest date for the work can be set well into the second century. The Church Fathers are fairly consisting in recognizing that the work was sometimes thought to be really St. Peter's, but that it was definitely not accepted as such universally; that is, some churches, but not all, read it as being genuinely Petrine. Origen, the first author to discuss its authorship directly, recognizes that some take it to be Petrine but does not himself regard it as such, and this is quite common among other Church Fathers who discuss these matters.
Thus the 'conservative' here can hold that (1) there is a connection to St. Peter, such that he was in some way involved in the formation of the epistle under his authority, but (2) this connection could be fairly loose, given the uncertainties and ambivalences that we find. Beyond the actual statements of the text and consistency with some patristic evidence, there is not much that can be put forward directly to support it. Some people note, correctly, that for a work that involves a self-attribution to St. Peter, it is quite restrained, in both theology and attribution; pseudepigraphy often attempts to call on the authority of apostles for very extravagant claims, but the epistle here does not seem to be doing so. It gives us a very minimal account of the Transfiguration, draws conclusions that are quite modest, and attempts to impose by apostolic authority things that are mostly also found elsewhere in the New Testament. How much weight one gives this argument depends on a large number of background assumptions.
The 'liberal' position would be the opposite, that the process of the epistle's formation involves not even a loose derivation from St. Peter and his authorization. This is often combined with, although it is not strictly required by the 'liberal' position on authorship, the view that the early apparent allusions to 2 Peter are actually the reverse -- that is, 2 Peter is influenced by those texts rather than vice versa. This would make its origin well into the second century.
When we look at direct arguments for 'liberal' positions (i.e., those that don't just point out the uncertainties of the reasons for the 'conservative' positions), we find, however, that they don't get us very far. People will commonly point to the later patristic doubts about the authorship of the work. But this is in many ways the strongest argument on the liberal side. Other arguments are consistently doubtful. The Greek of the epistle is very weird, and definitely different from 1 Peter, but taking this to mean that it is not Petrine at all requires making a number of assumptions that are only that, assumptions. The 'conservative' positions for 1 Peter and 2 Peter do not strictly require (although they don't necessarily rule out) that St. Peter himself directly wrote anything at all, only that the epistles were written and sent under his authority and on his behalf; and if we take a 'liberal' position on 1 Peter, one can't use stylistic differences with 1 Peter as reasons for thinking 2 Peter non-Petrine.
2 Peter shares a lot of material with Jude, and effort sometimes is made to parse what this might mean for authorship. The answer is that we don't know. We don't know for sure how the two are related. Jude is usually thought to be first, because it is much, much tighter in argument than 2 Peter, and seems to have a much closer familiarity with the book of Enoch on whose material it draws. What is more, 2 Peter's use of the material is somewhat odder, in ways that would be explained if the author had heard either Jude or Jude's original source, but didn't have a written copy at hand. (If we assume that 2 Peter is drawing from Jude, the author of 2 Peter at several points substitutes for expressions in Jude other expressions that are similar-sounding, suggesting that it was not from a written copy but either from hearing it or remembering it. If we are assuming that they are both drawing from the same third source, Jude's use is much more like what you would expect from someone who had an actual written copy, and 2 Peter's use is much more like what you would expect from someone who had only heard it or had it second-hand.) The argument, which one occasionally finds, that St. Peter would not draw on ideas by St. Jude, on the ground that St. Peter was more important than St. Jude, is childish and stupid, and marks the person who uses it as an idiot who can be dismissed. What is important, though, is that, assuming 2 Peter draws on Jude or at least Jude's source, which is plausible, we know, even on the 'conservative' position, that a significant amount of the material of the epistle is not actually from St. Peter himself -- at least, assuming (which we have no reason to believe but which we certainly do not know) that St. Peter himself is not Jude's source. (We can't fully rule that out because the ideas that someone cannot draw loosely on someone else's summary of what he himself previously said, or that someone cannot at times say things more precisely and at times more loosely, or that someone cannot remember what he previously said in a different way than actual record shows, are all obviously wrong.) The relationship with Jude, so far as we have it, doesn't really seem to say anything definite at all about the authorship of either.
Sometimes one finds arguments about St. Peter's literacy. He and the other apostles are referred to as agrammatoi in Acts 4:13, but that can mean anything from complete illiteracy to someone who can read or write at a basic level but has no literary skill. What is more, scribes were common in those days, and people could easily author letters despite not being able to write them. St. Peter is from Capernaum, which some think was not a Greek-speaking town; but a problem with this is that both Simon Peter and his brother Andrew have names that could be Helleno-Jewish -- that is, they have Jewish names that are very similar to Greek names, which seems to have been a common practice in Greek-speaking Jewish families in the day. And again, the 'conservative' position doesn't actually require that St. Peter directly wrote anything at all; he could have, like any number of other people, hired someone to write a letter on the lines that he determined. (The epistle doesn't show any signs of having been directly translated from another language, but, again, the 'conservative' position is consistent with St. Peter just asking for a letter that said such-and-such and the scribe writing the actual letter for him.) 1 Peter is generally thought to have fairly good Greek, but 2 Peter is very difficult to figure out in these terms, because it's usually thought to indicate wide reading in Greek (a lot of very artificial, literary flourishes) but is often baffling in ways consistent with the idea that the author does not speak Greek as a native language. On the basis of 1 Peter 5:12, the good Greek of 1 Peter is often attributed to Silvanus being the scribe. Recently there has been a push to interpret 1 Peter 5:12 as suggesting that Silvanus is the courier, not the scribe; the Greek, however, just says, 'I have written through Silvanus'. Literally, this would be a scribe of some sort; there are figurative uses of the phrase that could extend to other offices, including courier, but I have yet to come across any non-question-begging argument that it is ever used of someone who is only a courier. Thus a common position among 'conservatives' is that 1 Peter and 2 Peter are different just because Peter is using very different scribes at very different times and places, and this is possible, although we don't know.
If we assume that 2 Peter is pseudepigraphal, it is either deliberate pseudepigraphy (for instance, a rhetorical exercise) or a deliberate forgery. If the former, it would usually have indicators to indicate that it is intended to be taken as pseudepigraphical. Some people have argued that 3:2 is such (the author distinguishes himself from 'your apostles'); but again, 'apostle' in this period does not always indicate one of the Apostles proper. Forgery is always a possibility with any ancient text, but that is very different from our actually having evidence of it, which we don't here.
Thus, while the 'liberal' position is perhaps the easier position on the evidence, it is not by much. A great mistake people make in these kinds of disputes, whether about Biblical books or any ancient work, is assuming that the answer is in the evidence. In any case where there is extensive dispute at all, this is never true. Going on the evidence alone, some things are somewhat more probable than other things, but nothing can be established with certainty. No particular position is ruled out by the evidence. This sometimes happens. We do not always have complete evidence, and we certainly don't here. Our evidence is often consistent with many different opposing explanations, and this is certainly true here. The evidence can favor one position over another without getting anywhere near being definitive. Evidence can even be contradictory in a way that the evidence itself does not give us enough to resolve. And this, in the current state of things, seems very much the case here; it's background assumptions, not evidence, that get one to a definite conclusion.