How do I trap a mouse without killing it?

Try a humane mousetrap.
humane mousetrap - Google Product Search

Supposing you have an ethical reason for not killing mice, what do you plan to do with it after you catch it?

Check out a tin cat available at your local tsc store
It's a small steel box with one way door and a lid
Bait with peanut butter and wear gloves to avoid hantavirus "please read" Homepage | CDC Hantaviruses

^^ perhaps making dissections! =D