What is the difference in meaning between the nominative and the accusative?

The nominative is the case (the form of the noun) used to mark the subject of the verb.  That's the person or object (or anything) which is carrying out the action of the verb: the do-er.  In the sentence, 'The woman likes cheese': Die Frau mag Käse, 'the woman' (Die Frau) is in the nominative, as she is the one doing the action of liking the cheese.

The accusative is the case used to mark the object of the verb.  That's the person, or object (or anything), which is experiencing the action of the verb.  Take the sentence 'The dog bit the man': Der Hund biss den Mann.  Here, 'the dog' (Der Hund) is in the nominative, and 'the man' (den Mann) is in the accusative, because the man is the one to have experienced the bite.  This is really important to make clear in German, as mixing up the two would give you the meaning: 'the man bit the dog' (Der Mann biss den Hund).

(Optional background explanation)  Why does German have a system of marking the nominative and the accusative?  

German has a system of word order that is more 'free' than English, so we use the cases to show who is biting whom!  Germans can say 'der Hund biss den Mann' or 'den Mann biss der Hund' to mean 'the dog bit the man'.  The nominative and accusative cases, not the order in which you say 'man' and 'dog', tell you who bit whom.

We have the same system in English, although only in more formal language; 'who' is the nominative form, and 'whom' is the accusative form.  English used to have many more examples, but has developed to use word order, not cases, to make the meaning clear.