Learn Italian: Marco mi dà un libro. - Marco gives me a book.

In Italian, short pronouns like mi (which means "to me" here) usually go immediately before the conjugated verb. While in English we say "Marco gives me", Italian structures it as "Marco to me gives".

Does mi mean "me" or "to me" in this sentence?

Here it means "to me", because Marco is giving the book to you. This is called an indirect object. The pronoun mi is handy because it can mean both "me" (direct object) and "to me" (indirect object) depending on the verb.

Why is there an accent mark on the word dà?

The accent mark is just a spelling rule to tell it apart from the preposition da, which means "from" or "by". They sound exactly the same, but dà with the accent means "he gives" or "she gives".

Could I put the "to me" part at the end of the sentence instead?

Yes, but you wouldn't use mi. Instead, you would say Marco dà un libro a me. You use a me after the verb when you want to strongly emphasize that he is giving it to you and not to someone else. Otherwise, using mi before the verb is the most common way.