How many distinct Biblical canons are there?

As Richard answered, there is a difference between translations and canons. However, I wanted to add that there is also a difference in source text. This later subject is the matter of the field of Textual Criticism.

That is, nobody currently has the original autographs. What we have are copies, and often copies of copies, some of which come in fragments. For works written before the invention of the printing press, we don't have a guaranteed method of knowing the exact words of the original. However, we can make some very good guesses based upon the copies we see. There are different schools of thought on how to put these pieces together. For instance:

• Do we assume that the actual Bible was preserved by God to be available to the general public for all periods, or was it preserved, but in a hidden manuscript?

• Do you believe that monks making copies would be more likely to be lazy and leave a letter out, or more likely to add a letter to fit his doctrine?

  Because of this, there are different groups of texts which make up the different source texts. This may also inform your decision on what you believe canon to be.

As for canon, there have been completed lists of what is accepted as canon going back to at least the 2nd century. Some believe the Muratorian fragment to have been written in 170 AD. But the earliest lists either don't have a complete copy or there's disagreement with the authors of the lists.

However, the subject isn't quite muddied as much as you might think. We actually know more about the Bible than we know about almost any other work predating the printing press. We have many more fragments, in much more agreement than other work. Word for word, we know more about the words in the Bible than we do about the words of Shakespeare's plays. Also, I believe that every word of the New Testament has been quoted by the early church leaders in works which have more complete copies. Furthermore, at least for the most popular canons, the most important doctrinal themes are still present in the same way. So, while there is some disagreement, and even heated disagreement, the overlap is enough that the major doctrines can be agreed upon from them. For more information on the historical textual evidence, see Josh McDowell's Evidence That Demands A Verdict.

Each denomination has reasons why it believes the canon it chooses, and I have yet to see any confusion within a denomination on what it considers canon. For some, the canon has been so obvious that they never felt the need to spell it out. I don't think that the Catholics had to spell out the canon until the 1500s, which was probably done more as a side effect of another issue. The fact that you've never heard of other canons is itself testimony to how ubiquitous the standard canon is. If you were to take away all the books that people disagree about, you would have the standard Bible that most churches, at least outside of Catholicism, consider to be the full canon.

I'm going to ignore the part about the book of Mormon because I know absolutely nothing about it.

But as a short answer to a very big question, there are two ancient sources of scripture and the scripture tradition of all Christians are more or less built around them.

1. Septuagint

   The translation of the Hebrew Bible in Greek which is the source informing the earliest Bibles (which incidentally would be Catholic Bible and once the Great Schism happened, the Orthodox Bible).

2. Masoretic

   The Hebrew Old Testament, the canon of which apparently was approved at the council of Jamnia (a Jewish council which took place a few years after the Resurrection of Our Lord).

Moreover, you probably want to know which Bibles apply to which confessions. Well, the modern Catholic Bible which is approved by the Bishops relies on the Septuagint for its structure and canon (hence it is bigger). Translations of the Catholic Bible generally use both the Masoretic, Septuagint as well as the Dead Sea Scrolls for context and accuracy. But, approved translations for Mass (Like the New American Bible) reflect what's in the Latin (Vulgate) Bible (originally translated by St. Jerome in the 4th century). Protestants use the Masoretic text for the OT, but most translations include a bit of help from the dead sea scrolls I'd imagine (unless your a KJV only person). Orthodox on the other hand, use only the Septuagint.