VERY roughly (in my head, this isn't anything official):
Beginner (excluding 'never-ever' or someone who skied a couple times 20 years ago, etc.): reasonably comfortable on green trails, uneasy on blues, would have a lot of trouble on blacks, unable to deal with steeps/bumps at all. Can control speed on a green trail making linked wedge turns, or a mix of wedge turns and wedge christies.
Intermediate: very comfortable on green trails, reasonably comfortable on blues, some difficulty with groomed blacks. Could possibly deal with moguls on blue terrain, but would have a lot of trouble with 'black' moguls and a great deal of difficulty on 'double black' terrain. Able to control speed making linked wedge christie turns or a mix of christie/open parallel on groomed green/blue trails. Might revert to wedge or stem moves on steep groomed terrain or in moguls.
Advanced: no trouble with green/blue trails, reasonably comfortable on groomed black terrain. Can handle 'easy' bumps, probably some difficulty with black/double black moguls but able to survive them. Able to control speed making linked open parallel turns on any groomed green/blue/black trail, maybe with the occasional wedge christie or stem in more difficult terrain. Can possibly carve 'arc to arc' on easier trails.
If you're able to ski comfortably on ungroomed black/double black terrain or you can carve down a groomed DH course you're beyond 'Advanced' IMO.
I wouldn't put any specific age limits on those categories, although the part about speed control is important. A little kid who can 'power wedge' his way straight down a groomed black trail may tell you can he can "ski" it, but if he can't link turns down a blue while controlling speed he's a "beginner".
Some people will reach 'intermediate' very quickly, especially if they have skill crossover from an activity like skating or ice hockey. Most people are going to need some mileage (5-10 ski days?) to start feeling comfortable on blue terrain. Beyond that it's really variable and depends on how often you get out and how much you want to push yourself to improve.
EDIT: like Kevin said, there is definitely some variability in ratings between ski areas. It's certainly possible that, say, blue runs at a big mountain will be tougher than black runs at a small hill. And if you're used to skiing at a resort without much vert, skiing runs that are several times longer than the top-to-bottom vertical at your home mountain can be an eye-opening experience.