It's from the verb "tun", which means "to do" (and has the same root as the English "do"). It is used in compound verbs like "wehtun" (to hurt, takes the dative when transitive) or by itself just as it would in English. "Was tust du?" means "What are you doing?" In the case of "es tut mir leid", it's more idiomatic, since that literally means "It does sorry to me" or "it makes me sorry". The difference between "machen" and "tun" isn't totally clear, but I've noticed that "machen" is usually more emphatic or refers to doing something more specific (remember that "machen" can also mean "to make", while "tun" generally cannot).

"I am sorry for this/that" might is also a fitting as translation. As the "Es" expresses that something specific (this/that) was result of the situation that made yourself sorry. Your reflecting your own doing therefore using the reflexiv pronoun of I in its "Dativ" form "mir". Bram already gave a detaild answer to the use. I'll will give some more examples so. "tun" means that you, someone or something does some thing. Es tut sich etwas. Er tut Gutes. Niemand tut etwas. Es tut weh. Tust du das weg? there are alot more examples on http://www.duden.de/rechtschreibung/tun_handeln_erledigen in camparison you might look at "machen" http://www.duden.de/rechtschreibung/machen