Battery or Alternator problem?

I go to start my car by a boost. Once the car is started and boost cables
are taken off the car battery light comes on, revs up on its own till battery
light goes back off, then repeats the same process. Sometimes she even
shuts off.

However, when the car is still running and we take the battery posts off, the
alternator is able to run the car fine on its own. Many people tell me
different things from; battery problems, alternator problems, and even
wiring problems.

More information would be great!