What happens to diplomats who break the law?

If it's bad enough, they are expelled.

Nothing usually. They have diplomatic immunity wherever they are. That's an interesting article. It's up to the visiting country if they want to press charges and risk having an international incident with another country.

They have immunity during their tenure.

They become diplo-inmates.

1) "Foreign Ministry spokeswoman Cecilia Julin said foreign diplomats are only immune from criminal prosecution in countries where they have been accredited with the authorities.
"If you come to Sweden and commit a crime, you're just like any other foreign national," she said."
Source and further information:
http://news.yahoo.com/s/nm/20091120/od_nm/us_northkorea_odds
(article quoted in the question)


2) "Article 31

1. A diplomatic agent shall enjoy immunity from the criminal jurisdiction of the receiving State. He shall also enjoy immunity from its civil and administrative jurisdiction, except in the case of:

(a) A real action relating to private immovable property situated in the territory of the receiving State, unless he holds it on behalf of the sending State for the purposes of the mission;
(b) An action relating to succession in which the diplomatic agent is involved as executor, administrator, heir or legatee as a private person and not on behalf of the sending State;
(c) An action relating to any professional or commercial activity exercised by the diplomatic agent in the receiving State outside his official functions.

2. A diplomatic agent is not obliged to give evidence as a witness.

3. No measures of execution may be taken in respect of a diplomatic agent except in the cases coming under sub-paragraphs (a), (b) and (c) of paragraph 1 of this article, and provided that the measures concerned can be taken without infringing the inviolability of his person or of his residence.

4. The immunity of a diplomatic agent from the jurisdiction of the State does not exempt him from the jurisdiction of the sending State.

Article 32

1. The immunity from jurisdiction of diplomatic agents and of persons enjoying immunity under article 37 may be waived by the sending State.

2. Waiver must always be express.

3. The initiation of proceedings by a diplomatic agent or by a person enjoying immunity from jurisdiction under article 37 shall preclude him from invoking immunity from jurisdiction in respect of any counter-claim directly connected with the principal claim.

4. Waiver of immunity from jurisdiction in respect of civil or administrative proceedings shall not be held to imply waiver of immunity in respect of the execution of the judgement, for which a separate waiver shall be necessary."
Source and further information:
"Vienna Convention on Diplomatic Relations"
http://www.worldinbalance.net/intagreements/1961-vienna-diplomaticrelations.php

It seems, from what I've read and heard, that diplomats are, with few exceptions, pretty much "above the law."
I understand that the ones killing time at the U.N. have many hundreds of unpaid parking tickets, but are "immune" from any legal actions; they are aware of that, and take advantage of it.

Nothing because they are powerful people.

they get stuff, like a planetary gear.

If it's a serious crime, they are usually sent back to their originating country. If it's speeding or something like that, it's usually ignored.