An anime's soundtrack is usually made specific for the show. The same cannot always be said about the anime's opening or ending themes, though.
When it comes to producing an anime, multiple companies work together to make a show happen; these multiple companies make up what's called a Production Committee. Usually, a show's production committee will include a music label (such as Aniplex or Lantis), and this label can provide a composer to make music for the anime. There are cases and times, though, where a specific composer can be brought in from a certain label (such as Yuki Kajiura being brought in to work on Madoka Magicka), if they have enough reputation built up around them.
Music production is generally done with a computer and a digital piano to create all the different instruments. Maybe some other instruments are brought in, such as brass instruments, to give the live instrument feel, but generally, a computer can synthesize another instrument's sound fairly well. It's also a lot easier than trying to find and organize (and pay, most importantly) a music group or symphony to play and record all of a show's background music. (Again, not to say that it isn't done, but it's not a general thing.)
With the vocal stuff in the background music, I couldn't specifically say how that gets managed, but my guess would be that they get vocalists from the composer's music label to sit in and sing the parts. Like many things in anime, voice acting is definitely a competitive field, so for vocalists, being able to get at least some job singing for background music is better than nothing.
Finally, to answer your question about how they decide the feel of the sound, it's generally hashed out at meetings early on in an anime's creation, with the composer (or someone representing his/her label) there along with the anime's production staff. Depending upon how much the production staff knows about music, the composer could get very specific or very vague descriptions of the sound they're looking for. Usually the words thrown around are adjectives that depict a certain feeling (like "dark, scary music" or "light, airy, cheerful music"), unless, again, the production staff know more about music and can give more music-focused terms.
When it comes to how the music itself is created, and how composers get the training to get their jobs, just look up "music composition" and "digital music composition" on Google and it'll start to point you in that direction. A lot of the same general composition and creation processes apply, although I can't really tell you if they do something specifically unique in Japan composition-wise. Anyway, people can go to college to study music theory and music composition, and from there, they may choose to work specifically for anime, or choose to work anywhere else (like composing for live-action TV, or for movies, or commercials, or for the entertainment industry, for example). It's a career and profession like any other.