An appellate litigator writes briefs and does oral arguments. And that's basically it; there's very little client interaction, no fact-gathering at all, and very little in the way of motions. What this means, in practice, is that an appellate litigator spends most of his or her time reading judicial opinions deciding cases, reading statutes, reading regulations — basically,
reading law — and writing about how the law applies (or ought to apply) in some particular case. Every few months, after you've filed your briefs, you go to court and judges ask you (hostile) questions about your position and why it makes sense.
Many law firms have appellate groups, and a lot of government bodies (the federal government, some agencies, some states) have appellate specialists. In a law firm, it's likely that you are a general brief-writer, which means that you may be brought into a case at the trial court level in order to write a brief regarding a motion for summary judgment or some other dispositive motion; this is not terribly different from an appellate brief, but it's a different location in the process, so it looks a little different. If you're a government appellate litigator, you probably just work on appeals (no dispositive motions in trial courts).
Most appellate litigators are in DC. In a law firm with multiple offices, the DC office has most, if not all, of the appellate staff. In the federal government, the appellate staff is in DC. States will usually locate their appellate litigators in the state capital, although there are some exceptions (e.g., California).
This practice area is
extremely hard to get into. You really have to go to a top school (probably no lower than CCN, and HYS would be better) and be near the top of your class. Being on the flagship law review is a major plus, and some kind of journal work is basically a requirement. Doing (very) well in a moot court competition is also a plus. Also, you have to have a judicial clerkship
after law school; there's basically no other way (although I think I know of one exception among my friends). A federal appellate clerkship is probably required, and a feeder judge helps a lot. A Supreme Court clerkship helps even more, but that's even harder to get. (To put it in perspective, about 1% of HLS gets a Supreme Court clerkship.)
As to your specific questions, yes, one can become an
appellate lawyer right
after law school, although probably you need a clerkship (or two) to get you started. The pay is the same as for anything else; firm will start you at 160K and move you up the usual pay scale. Governments will pay substantially less.
On law firm appellate practice: Attracting a sufficient quantity of work to hit your hours for the year and be considered for partner is really hard in this field, because you need to get substantive experience (i.e., actually argue cases), and in law firms, the most senior person on the case is the one who does that. So the only way to get that experience, really, is to work for the government. So expect to leave the firm and get a government job for a few years, at least.
Oh, and I'll add that appellate practice is really, really fun if you like law and reading and writing, mostly by yourself. I'm doing an appellate internship right now, and I like it a lot.
EDIT: Totally forgot about criminal appellate. See below.