/clears throat/ Okay, let's do this.
Essentially, your basic explanation is pretty accurate. The Babylonian Talmud was the one written by the Jews who were exiled to Babylon; the Jerusalem Talmud was not actually written in Jerusalem, but it was written by the Jews who stayed in Israel (it's just that they mostly weren't in Jerusalem by that point, but rather in varying places across the country).
The Talmud itself is actually an expansion of the Mishnah, which was the written-down version of the oral tradition/oral Torah; Rabbi Judah HaNasi codified the Mishnah due to the rising fear that all the rabbis would be killed, and the tradition would be lost. However, the oral Torah was actually not supposed to be written down...so he wrote it in as few words as possible. This caused much confusion for later generations.
The Talmuds were later written in order to a, figure out exactly what the Mishnah's talking about, and b, to discuss such things as whether the law must be updated for the current time. Both were written over quite a number of centuries and generations, which are actually divided into at least five or six categories (with the modern-day scholars being part of the last one). They feature different people, slightly different styles of Aramaic, and one will occasionally cover a topic from the Mishnah that the other doesn't (most notably the topic of laws that only apply in Israel). The Talmud also provides insight into the people involved in the conversations it records (the coolest description I've ever heard of it was something like 'the Talmud makes it sound like all the rabbis of various time periods and places are sitting together and conversing.'
This, of course, is occasionally coupled with stories, which are generally meant to teach you something but also sometimes say funny or interesting things about the rabbis in the stories). I don't entirely know how it is in other schools, but in my high school Talmud classes we ended up with a lot of laughter and many inside jokes. (I don't mean to imply that we didn't learn a lot, because we did; we just had a whole lot of fun doing it).
The Talmud (both of them) follows the structure of the Mishnah, and is divided into sections ("orders"), which are then divided into smaller sections ("tractates"). Each tractate can easily span 1-3 volumes, although they admittedly come published with commentary; I believe there are 62 tractates in total, although my numbers could be off. Either way, it's very, very long. (Tanakh, Extended Edition indeed!) The Talmud as a whole is, along with the Mishnah and, of course, the Tanakh, one of the most important books in Judaism, even today. Why? Because whatever law you can possibly be unsure of, it's been discussed and recorded in there. And even if it hasn't, there'll be something similar that can help you out.
Due to the Talmud's length, there have been a whole lot of people who made halakhah/law books--sort of 'Talmud: Condensed Edition', if you will. The lengthier of these are usually just going through the Talmud--yes, all of it--and recording only the decisions. Did I say 'just'? Ha. No. It's a huge job, and probably took years.
But why, you ask, would anyone have to record the decisions? Surely they'd only have to filter out the stories?
Well, no. I mean, yes, a lot of the stories don't contain rulings on law. However, you remember when I mentioned conversations earlier? Yup. That's what the majority of the Talmud is: two rabbis have an argument. One says that tree branches cannot be collected on a holiday under any circumstances; the other says that if they were already on the ground, it's fine. This debate is recorded. A century later, another rabbi says to his study partner that branches can only be collected from the ground if it's certain that they fell there that morning; his study partner tells him that that's ridiculous, because the proof that the original second rabbi brought in (a hundred years ago) clearly states that the branches fell and were collected--that is, immediately collected. Fifty years later, one student quotes this debate to his study partner, who promptly bangs his fist on the table and brings in proof that says the original proof only applies on weekdays, thereby negating it. Their teacher then comes over, and says, "What, do you really think that's right? Impossible! I have further proof that means that the first proof still works, and also negates the one you just brought in. The law is with the original second rabbi, done." And then they probably argue over it for a while, but the Talmud only really records the boiled-down message of what each one says, and so the extra arguing isn't recorded. The Talmud then states that the teacher is right, the law is with the original second rabbi, and the case is closed.**
(Proof here is sentences from the Tanakh that indicate law or that can be used in conjunction with other sentences to indicate law. For example, in a discussion on whether or not girls should be taught Torah, the quote "and you shall teach your sons" is often brought up, with the addendum '--your sons, and not your daughters'. That particular debate, however, has continued through many centuries and many opinions, and has reached the point where a number of rulings say that it's perfectly fine for girls to study Torah, Talmud, and all the rest of it. This is what enables me to be sitting here and writing about my high school Talmud class; while the school is a, mixed gender, and b, has students from most of the religious denominations of Judaism, they would definitely have received backlash from part of the community if we were still living, say, in Talmudic times. Or several hundred years ago. Have you ever seen Yentl? There were certainly exceptions, but for the most part girls were limited to lessons from the Torah.)
Mind you, these conversations were not recorded at the time they happened. Instead, they were passed down through memorization.
That's right. For years, people memorized the Talmud--yes, all of it--and taught it by reciting it. Presumably, the new debates were added in as they arose. (Obviously, no one person memorized the whole thing, but people would know an entire book. Or two. It makes studying it today in written form sound so simple.) I don't currently remember when (or why) it was written down, though I assume the 'why' is yet again because they were afraid it would be lost. As the Talmud is the Jewish people's most major book of law, losing it would be a great tragedy.
Mind you, not everything that Jews do today--or even in the past--is written in the Talmud. A whole lot of things arose through tradition, which was then accepted into the general religious culture; one rabbi actually states in the Talmud that traditions can become the halakhah (or law) simply by virtue of being so widely accepted. (This is as far as I remember; there is a possibility that he or another person then negated that. I'm pretty sure someone said that traditions can occasionally or often become law, though, so if I'm wrong...well, my bad. Sorry).
Anyway. While I could quite easily continue with this explanation of the Talmud, this does cover the major details, and is probably more than anyone was looking for. I'm quite happy to make a sequel, though (which would include notes on the structure of the pages, and on who commentated, and what some of the stories are, and all of that).
(If anyone's curious: I studied Mishnah for...a year, I think, and Talmud for four. Well, technically it was six, but the classes got more serious in high school. I've taken shorter courses since then, and have covered the history of the Talmud and the Mishnah and so on in Jewish History classes for more years than I'd really care to try to count just now, including last year in a university course. So. That would be my credentials.)
By the way, hS, where on earth did you get your knowledge? It's extremely accurate, in both details and terminology.
~DawnFire, Jewish Studies major, signing off
PS: Word count says this is over 1,000 words. If you read it all, I commend you and hope it was interesting.
PPS: Forgot to mention--the Mishnah is written in Hebrew, while the Talmud essentially takes a sentence from the Mishnah and then records the debates and so on in Aramaic. The commentators come in very handy while studying it these days, as does a teacher. And the occasional dictionary or translation.
(Or, y'know, an Aramaic class. I want to take one at some point. Apparently, one of the universities in my city offers one. But it's certainly possible to study Talmud without it! Why, I'm living proof!)
/backs away from the computer and goes to make tea/
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**I should mention that this case/exchange is completely made up; there is, however, at least one dealing with what you can do with branches that have fallen on a holiday, including what you can do if they've fallen of their volition into your outdoor oven. I think it's in Masechet Beitzah (aka, Tractate Egg).
(It takes its name from the first word in the tractate, which is, of course, 'egg'. Specifically, 'an egg that was laid on a holiday, the house of Shammai says it may be eaten, the house of Hillel says it may not be eaten.' That was a fun tractate. Apparently, one of the physics teachers actually came into the other class to show them how an egg's structure meant that it actually could be used to prop up a bed--or in this case, a desk--without breaking. And yes, that's one of the purposes of eggs being discussed in the tractate. Pretty sure we were all incredulous when we first read that.)