The discussion of what words (or sentences) mean is called "semantics", as opposed to the discussion of what signs mean (semiotics).
A. Prosodic Meaning: The way a sentence is said can radically alter the meaning. Emphasis can make implications on what is taken for granted, and what is of special significance:
Where were you last night?: request for location
Where were you last night?: emotive statement
Where were you last night?: as opposed to anyone else
Where were you last night?: as opposed to any other night
Where were you last night?: as opposed to any other time
B. Grammatical Meaning: "John read a book yesterday": you could look at this as Subject + Verb + Object + Adverbial (grammar), but you could also deal with it semantically: an actor performing an action on a goal at a certain time.
The very structure of our language provides the possibility for certain types of meaning, and blocks others off. We have an "active", subject/object oriented language. People (subjects) act (verb) on things (object). Not every language is like this. In English, word order conveys meaning; in other languages, not at all.
C. Pragmatic Meaning: The function performed by a sentence also has to be taken into account. The meaning of "I'm hungry" seems plain enough, but it could sometimes be a statement of fact, and sometimes be a veiled command ("Feed me").
D. Social Meaning: The choice of a sentence may affect social relationships between conversants. We can convey politeness, rudeness, competence, or other things which affect our status or role in a community. "What do you mean talking to me like that?" raises more than the lexemes indicate.
E. Propositional Meaning: Propositions are logical units; sentences are grammatical units. "Joe must have a cold, because he's sniffling a lot and talking funny." is one sentence, but two propositions. Meaning can be taken as logical structure. Propositions are units of language to which we can ascribe a truth value (that is, we can say that the claim is true or false).
This is by no means an exhaustive list of ways in which we determine the meaning of a sentence.