Cognitivism is the meta-ethical view that ethical sentences express propositions and can therefore be true or false (they are truth-apt), which noncognitivists deny. Cognitivism is so broad a thesis that it encompasses (among other views) moral realism (which claims that ethical sentences express propositions about mind-independent facts of the world), ethical subjectivism (which claims that ethical sentences express propositions about peoples' attitudes or opinions), and error theory (which claims that ethical sentences express propositions, but that they are all false, whatever their nature).
Propositions are what meaningful declarative sentences (but not interrogative or imperative sentences) are supposed to express. Different sentences, in different languages, can express the same proposition: "snow is white" and "Schnee ist weiß" (in German) both express the proposition that snow is white. A common belief among philosophers who use this jargon is that propositions, properly speaking, are what are true or false (what bear truth values; they are truthbearers).
To get a better idea of what it means to express a proposition, compare this to something that does not express a proposition. Suppose someone minding a convenience store sees a thief pick up a candy bar and run. The storekeeper manages to exclaim, "Hey!" In this case, "Hey!" does not express a proposition. Among the things that the exclamation does not express are, "that's a thief there"; "thieving is wrong"; "please stop that thief"; or "that thief really annoys me." The storekeeper isn't saying anything that can be true or false. So it is not a proposition that the storekeeper is expressing. Perhaps it is an emotional state that is being expressed. The storekeeper is surprised and angered, and expresses those feelings by saying, "Hey!"
Ethical cognitivists hold that ethical sentences do express propositions: that it can be true or false, for example, that Mary is a good person, or that stealing and lying are always wrong.