I took a class in Kierkegaard and it was represented that Kierkegaard's use of pseudonyms was highly significant. Either he meant what he said, or he was being ironic, or something else entirely. Is there an academic consensus forming around the meaning behind the pseudonyms, or were the pseudonyms just something people did in those times, and we can take the pseudonymous writings as though they were signed by Kierkegaard himself?
Well, there is a scholarly consensus that the use of pseudonyms was highly significant; I don't know of any Kierkegaard scholars who argue that it isn't meaningful.
Here's Kierkegaard himself on the subject:
"... As is well-known, my authorship has two parts: one pseudonymous and the other signed. The pseudonymous writers are poetic creations, poetically maintained so that everything they say is in character with their poetized individualized personalities; sometimes I have carefully explained in a signed preface my own interpretation of what the pseudonym said. Anyone with just a fragment of common sense will perceive that it would be ludicrously confusing to attribute to me everything the poetized characters say. Nevertheless, to be on the safe side, I have expressly urged that anyone who quotes something from the pseudonyms will not attribute the quotation to me (see my postscript to Concluding Postscript). It is easy to see that anyone wanting to have a literary lark merely needs to take some verbatim quotations from "The Seducer," then from Johannes Climacus, then from me, etc., print them together as if they were all my words, show how they contradict each other, and create a very chaotic impression, as if the author were a kind of lunatic. Hurrah! That can be done. In my opinion anyone who exploits the poetic in me by quoting the writings in a confusing way is more or less a charlatan or a literary toper."
As for the particular meanings of the various pseudonyms, and how they relate to each other and to Kierkegaard's own opinion: this is a matter of much scholarly discussion, and I don't think there is any solid consensus. I remember reading (in the late 1970s) a good article by Alastair McKinnon on the topic, but I imagine the field has shifted quite a bit since then.
There's three answers in the literature with respect to this question.
First, there are certain postmodern readers (I'm thinking of a Derrida text but could be mistaken) who take the pseudonyms very seriously and think there's no author behind them (since they accept the death of the author as a matter of semiotics and structuralism). The thing they like is that pseudonyms make this more obvious. I don't think there's much more worth saying on this (consider the following: if they are right, then every written work by everyone is pseudonymous; if they are wrong, then the view doesn't need further treatment).
Second, there's a group of scholars who think we can attribute everything to Kierkegaard himself. This group includes mostly non-specialists in Kierkegaard who do so just by accident. Such authors tend to attribute really strange things that the esthete says in Either/Or to Kierkegaard. But it also include figures like Jon Stewart who defend the idea that Kierkegaard's pseudonyms are just part of a common practice of using pseudonyms where everyone knows who you are that was common practice in Denmark at the time (see Stewart's Kierkegaard's relation to Hegel Reconsidered for more on this view).
Third, there's Kierkegaard scholars and others who take Kierkegaard at his word (from Point of View as an Author and also some journal entries). This view is propagated by the Hongs (translators of the Princeton series of Kierkegaard texts), Merold Westphal, C. Stephen Evans, Cynthia Walsh, et al (see for instance). This seems like a good idea but can be done naively and intelligently. There are several ideas and themes that are repeated across pseudonymous and eponymous books (or across multiple pseudonyms). There are also editing issues where Kierkegaard had planned to release in his own name and only hastily switched to pseudonym. These issues are sometimes pertinent in that the voice of the text is still written as if it is Kierkegaard even when it is not.
The recent book Kierkegaard: A Single Life sheds some more light on his usage of pseudonyms. Their usage seemed to be common in the day - as a means to distance oneself from what could be rough edges.
Even Bishop Mynster had a pseudonym named "Kts", made with the middle letters of his name (JaKob PeTer MynSter).
Copenhagen was a small place - it was a way for people to say what they really thought, but still be able to appear in polite society. No matter that people could still guess who was behind some pseudonyms.
Kierkegaard, as far as I know, was unique in having multiple pseudonyms that would interact and sometimes contradict each other. It seems clear to me that central to K's philosophy was the different possibilities of living a life, and it seems appropriate that K would explore those different possibilities through different personalities who have different ways of living their lives (most easily seen in the two parts of Either/Or).
This is intended primarily as a supplement to Michael Dorfman's excellent, primary-sourced answer
Similar to Plato, who rarely if ever wrote in his own voice, Kierkegaard had a philosophy deeply opposed to didacticism, and was endeavoring to convey concepts that were often paradoxical, or that couldn't be expressed directly.
Therefore, just as in Plato's dialogues, Kierkegaard presents a range of characters, each with their own philosophies, as stages along the way towards a perfected philosophy that cannot be directly expressed. The characters themselves can be didactic, but Kierkegaard is disclaiming endorsement of their dogmaticism, he is merely presenting them as objects of study (or at least, so he can claim).