Generally, the accusative object is the direct object, whereas a dative object is an indirect object. In the sentence 'I give the book to the man' the object on which the action is directly performed is 'the book', and that would be in accusative case, whereas the man is not acted upon but is the recipient of the action, which means he would be in dative case if English had a case system like Russian or German.

An example: in Russian you say дай мне ..., 'give me ...'. Мне is dative. The person saying this wants to get something. Дай меня ... would mean 'give me (e.g.to someone)', so the person speaking is not the recipient but what is being given.

Robin