Well it all depends on what level you produce on. A Producer in a local live show can do just about everything except for host the actual show. You will screen calls, you will book guests, you will sometimes do Prep work for info for the host to use on the show. Then if you ever make it to the Network level, or produce a syndicated show your role will be more specific. You could be the Executive producer if you so desire. This position would oversee the rest of the production team. Bigger shows usually have separate people screen calls, doing the prep work, and even writing material for the host to use. The Exec Producer is usually the one to book guests for the program as well.
I have produced programs on the "small market local scene," all the way up to Network Level Programs on both Fox, and ESPN Radio.
A degree in Communications (Radio and TV) is very help full for this position. Some of the smaller market stations will not even demand a degree. The will hire you as a Board-Operator (another basic name for a producer) to gain experience. then move you up the ladder.
If you are looking to break into the business, and need any pointers, you can feel free to email me, and I will try to point you in the right direction.