My son is the tenant in an apartment. He has several roommates. One roommate is making life difficult for the rest by making a mess, not cleaning up and by being in a foul, negative mood all the time. My son has had enough and told the roommate to move out, giving him 3 weeks notice. Is my son within his rights? There is no contract signed. Can the roommate sue him. The roommate paid part of the last month rent, can this be used for his last month?

Your son or the landlord needs to give him a written letter saying he has to leave. Then if he does not leave your son or the landlord needs to take him to court.
Verbal only he can get sued.

Who does he rent from, does he pay your son the money or does he pay directly to the owner.

If your son rented to him then your son can evict him. If the owner rented to them, the owner has to act.

He also normally has to be given a 30 day notice unless he is evicted for non payment

If your son is the only legal tenant (he's on a the lease with the owner), then any roommates he has are HIS tenants. If there is no lease between him and his tenants then they are month to month tenants. Depending on where the apartment is, dictates how much notice he has to give. Generally 3 weeks is not enough.