Commissions, what theaters give them out?

As I understand it, there are two categories. One would be regional theatres (call these semi-pro). If they decided to produce your work, they will pay a per-performance fee (not much, and they varry). The more times the play is put-on, the more you make.
Should you be fortunate to be picked up by the professional circuit (Broadway and all the off-Broadway companies as example), the pay goes up. Same deal, if you run for years (like Cats, Hair, Our Town, Death of a Salesman etc.) you'll do well.
Another avenue is getting listed on the internet and offer your work for a one time production fee. Schools, churches, local groups, etc.are always looking.
If any of this is incorrect, someone correct me.
Don't give up your day job!

I don't know how much use this is but if you are planning on applying to UK theatres there is a set fee set out by the Writer's Guild and the ITC (Independent Theatre Committee) which almost all professional theatres big and small adhere to, though there is a separate agreement for the big London theatres and long run shows. Either check theatres on-line or through something like the Writers Handbook for submission guidelines. Almost all theatres accept unsolicited enquiries though their submission guidelines will vary. Some specialize in new writing and most theatres will have a literary manager and/or a team of readers and rarely charge a reading fee or for a script report. In addition to this a good number of theatres run competitions to find new talent (Kings Cross/Soho) and the prize usually includes a full commission and performance though this could be for shorter plays as well as full length. Of course you can try amateur dramatics, schools etc but don't be afraid to approach professional theatres though it is a good idea to familiarise yourself with the type of work they produce to see if your play will be a good fit for them. A good starting point can be local theatres as well as new writing festivals but a little research will unearth a wealth of theatres looking for and producing new writing.

Good luck.

I'm uncertain what you mean by commission. The literal definition of the term is to pay a playwright in advance for a play the theatre wishes him/her to write, according to their terms, guidelines, or an agreement made beforehand. Is this what you mean? It's rare a theatre enters this arrangement with a playwright they have not worked with extensively. The best way to get commissioned to write a play is to have completed plays produced widely and cultivate relationships with those theatres. Then you might approach them with an idea that you might tailor to them, or they may ask you specifically if they're looking for a certain thing.

If you mean royalties, as in being paid per production from your work, than it varies. Most theatres are prepared to pay at least some royalties, even for new productions, unless production is the award for a contest or something like that. You'd have to inquire with each theatre individually. Many theatres are up for negotiation in this regard, and don't necessarily have a set rate.