Learn German: Es tut mir sehr leid.

In English, you say 'I am sorry', making 'I' the subject. But in German, the phrase is literally 'It does sorrow to me'. Because the sorrow is happening to you, you are the receiver of the action. Therefore, you use the dative pronoun mir (to me) instead of the subject pronoun ich (I).

What does the verb tut mean in this sentence?

tut comes from the verb tun, which means 'to do'. So the sentence literally translates to 'It does to me very sorrow'.

Why is leid at the very end of the sentence?

The phrase 'to be sorry' in German is based on a separable verb: leidtun (literally 'to do sorrow'). When you use a separable verb in a normal sentence, the main part of the verb (tut) takes the second position, and the prefix (leid) gets kicked all the way to the end.

How would I say 'He is sorry' or 'We are sorry'?

You just change the dative pronoun! The 'It does sorrow' part (Es tut ... leid) stays exactly the same. For 'He is sorry' (It does sorrow to him), you would use the dative pronoun for 'he' and say Es tut ihm leid. For 'We are sorry', you would say Es tut uns leid.