A premise is a proposition that tries to prove something. A conclusion is the proposition that the premise tries to prove. The simplest argument contains one premise and once conclusion. Therefore that hairy thing is probably a dog. The first sentence is the premise proposition, and the second is the conclusion proposition.
Most arguments have more than one premise.
All men are mortal, and Socrates is a man. Therefore, Socrates is mortal. In this case, the first sentence contains two premises, and the last sentence contains the conclusion. The conclusion does not have to come after the premises, however. Often, you will see the conclusion stated first, and the reasons given later.