As I noted in a previous column, American Sign Language derives from French Sign Language. According to Ethnologue, about fifteen years ago there were between 100,000 and 500,000 speakers of ASL in the US, putting it behind only English, Spanish, Cajun French, Hawai'i Creole English, probably Romani, and possibly Russian in terms of number of speakers in the US. (Ethnologue doesn't list Arabic and Chinese speakers for the US, though, so those may also have more US speakers.)
ASL is a natural language, developed over time by native speakers. In this regard it's unlike Signed English, the other major sign system for the deaf in the US, which isn't a language at all but rather an encoding of English. ASL has its own grammar, which doesn't match that of English. For example, there's a book by a hearing woman raised by deaf parents called Train Go Sorry: Inside a Deaf World; the title is a literal translation of an ASL phrase that more loosely translates to "You missed your chance." In ASL, the phrase is a perfectly grammatical sentence, whereas a Signed English equivalent might consist of the signs for you, missed, the, and boat. (My understanding is that ASL is easier for deaf people to learn and to understand than Signed English or written English, but I'm not sure how strong the evidence is. One case study is available on the Web.)
Of course, there's more to ASL grammar than just word sequence. My understanding is that inflection is handled largely by facial expression—to make a phrase into a question, for instance, you put on a questioning look as you sign the phrase. When I took an extracurricular basic-ASL class during college, that was one of the things I tripped over; when I concentrate hard my face tends to go blank.
The other major difficulty I had was in remembering the signs. My memory is more symbolic than visual or somatic; I think of English words in terms of their spellings, more or less. But I had no written symbolic system with which to represent sign vocabulary, so I had nothing but mnemonics to fall back on. I never got very good at it.
One thing I did learn was fingerspelling, also known as the manual alphabet. It's nice to know the letters, but I'm still not very fluent in them. And reading the fingerspelling letters is even slower for me than using them—it takes me about a second to recognize each letter, and anyone who's half-decent at fingerspelling goes at several times that speed, so I spend a lot of time being puzzled.
(I find it interesting that the word-signs are not composed of combinations of the letter-signs. (Although often a word-sign incorporates the initial letter of the word.) It reminds me a little of Japanese.)
The fingerspelling letters do come in handy when choosing a sign name. At least in some parts of the US, one's sign name consists of one's first initial used in a sign. So someone named Alison who's a teacher might have a sign-name consisting of the sign for teacher, made with a hand forming a fingerspelling A. (That particular example might not be feasible, I'm not sure.)
I always thought ASL would be really handy to know, even for hearing people: to communicate with deaf people, sure, but also for communicating in noisy restaurants, or from one street corner to another across a busy street. Lip-reading would be a similarly useful skill (and would have the added advantage of letting you eavesdrop on other people's conversations), but it's probably harder to do accurately from a distance. And ASL can be used for other purposes as well. I've attended several concerts, for instance, in which ASL interpreters signed the songs; for those who haven't seen this done, it's somewhere between simultaneous translation and interpretive dance. Some of the nuances of the English lyrics get lost, as in any translation, but as with other kinds of translation, a real artist can produce translations which are beautiful in their own right. Even to people like me who are illiterate in ASL; I've found I can learn bits and pieces of ASL by watching such interpreters, much as I can learn tidbits of other vocal languages by listening while reading subtitles in foreign films.
Unfortunately, "subtitles" in ASL pretty much require a person acting as an interpreter, because there's no good written form of the language. I've seen books that attempt to teach ASL vocabulary using photos and diagrams; they always seem inadequate, and the curved arrows don't really show me how to move my hands. I've also seen at least one attempt, years ago, to create a written ASL using stylized diagrams; the printed "words" were big and ugly (though that may have been because they were printed on a dot-matrix printer). And of course you can transcribe ASL into written English words, but then you have to decide whether to retain ASL grammar or not.
One promising approach, still in its infancy, is animated 3D "avatars," computer models of humans, which can be programmed to sign. For example, the SignSynth program allows you to type text and have a signing avatar fingerspell it. This technology has a long way to go—among other things, it uses VRML, which is far from ubiquitous, so you'll have to install a browser plugin to see it, and it still tends to be slow and a little clunky, and of course computer models are not yet as expressive as humans. But some day it may be possible to reduce the cost of ASL interpretation by having a human do the initial translation (into written English words in ASL grammar), and then having an avatar do the signing.