Overview
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. Cognitivists believe that these sentences do not just express feelings, as though we were saying, "Hey!" or "Yay for Mary!"; they actually express propositions that can be true or false. Derivatively, a cognitivist or a realist would say that ethical sentences themselves are either true or false. Conversely, if one believes that sentences like "Mary is a good person" cannot be either true or false, then one is a non-cognitivist.
Read more about this topic: Cognitivism (ethics)