You would expect an expert in semantics to know a lot about what meanings are. But linguists haven't directly answered this question very successfully. This may seem like bad news for semantics, but it is actually not that uncommon for the basic concepts of a successful science to remain problematic: a physicist will probably have trouble telling you what time is. The nature of meaning, and the nature of time, are foundational questions that are debated by philosophers.
We can simplify the problem a little by saying that, whatever meanings are, we are interested in literal meaning. Often, much more than the meaning of a sentence is conveyed when someone uses it. Suppose that Carol says ‘I have to study’ in answer to ‘Can you go to the movies tonight?’. She means that she has to study that night, and that this is a reason why she can't go to the movies. But the sentence she used literally means only that she has to study. Nonliteral meanings are studied in pragmatics, an area of linguistics that deals with discourse and contextual effects.
But what is a literal meaning? There are four sorts of answers: (1) you can dodge the question, or (2) appeal to usage, or (3) appeal to psychology, or (4) treat meanings as real objects.
(1) The first idea would involve trying to reconstruct semantics so that it can be done without actually referring to meanings. It turns out to be hard to do this -- at least, if you want a theory that does what linguistic semanticists would like a theory to do. But the idea was popular earlier in the twentieth century, especially in the 1940s and 1950s, and has been revived several times since then, because many philosophers would prefer to do without meanings if at all possible. But these attempts tend to ignore the linguistic requirements, and for various technical reasons have not been very successful.
(2) When an English speaker says ‘It's raining’ and a French speaker says ‘Il pleut’ you can say that there is a common pattern of usage here. But no one really knows how to characterize what the two utterances have in common without somehow invoking a common meaning. (In this case, the meaning that it's raining.) So this idea doesn't seem to really explain what meanings are.
(3) Here, you would try to explain meanings as ideas. This is an old idea, and is still popular; nowadays, it takes the form of developing an artificial language that is supposed to capture the "inner cognitive representations" of an ideal thinking and speaking agent. The problem with this approach is that the methods of contemporary psychology don't provide much help in telling us in general what these inner representations are like. This idea doesn't seem yet to lead to a methodology that can produce a workable semantic theory.
(4) If you say that the meaning of ‘Mars’ is a certain planet, at least you have a meaning relation that you can come to grips with. There is the word ‘Mars’ on the one hand, and on the other hand there is this big ball of matter circling around the sun. This clarity is good, but it is hard to see how you could cover all of language this way. It doesn't help us very much in saying what sentences mean, for instance. And what about the other meaning of ‘Mars’? Do we have to believe in the Roman god to say that ‘Mars’ is meaningful? And what about ‘the largest number’?
The approach that most semanticists endorse is a combination of (1) and (4). Using techniques similar to those used by mathematicians, you can build up a complex universe of abstract objects that can serve as meanings (or denotations) of various sorts of linguistic expressions. Since sentences can be either true or false, the meanings of sentences usually involve the two truth values true and false. You can make up artificial languages for talking about these objects; some semanticists claim that these languages can be used to capture inner cognitive representations. If so, this would also incorporate elements of (3), the psychological approach to meanings. Finally, by restricting your attention to selected parts of natural language, you can often avoid hard questions about what meanings in general are. This is why this approach to some extent dodges the general question of what meanings are. The hope would be, however, that as more linguistic constructions are covered, better and more adequate representations of meaning would emerge.
Though "truth values" may seem artificial as components of meaning, they are very handy in talking about the meaning of things like negation; the semantic rule for negative sentences says that their meanings are like that of the corresponding positive sentences, except that the truth value is switched, false for true and true for false. ‘It isn't raining’ is true if ‘It is raining’ is false, and false if ‘It is raining’ is true.
Truth values also provide a connection to validity and to valid reasoning. (It is valid to infer a sentence S2 from S1 in case S1 couldn't