Cup and cone is the older type of BB. This type has cups that screw into the threaded ends of the bottom bracket shell. The cup is a circular race in which the bearings run. Picture something like a shallow bowl with the center cut out. The axle or spindle has cones on each end, spaced so that they trap the bearings in the cups. The cup on the (usually) drive side is called the fixed cup or fixed race. It has to be very tight. Some people suggest that one let a shop install this type of bb because it is difficult to get the fixed cup tight enough without special tools. At home one generally does not mess with the fixed cup. The cup on the non-drive side is called the adjustable cup. It is held in place by a lockring that usually has notches. One uses a special bb lockring wrench that looks like a hook with 2 or 3 "teeth" that engage the notches.

To service such a bb, one uses the lockring wrench to unscrew the lockring, unscrew the adjustable cup, and pull the spindle out (you have to remove the cranks first), catching the bearings on a towel cleverly placed under the bike BEFORE starting the process. The bearings may be loose or in a retainer.

The other type of bb is sealed cartridge. The bearings are sealed in units that screw into the bb shell and are effectively both "fixed" since they are basically screwed in as far as they will go.

For more you really need to get a book. Hope this helps.
Regards,
Raymond