Generally the word きれい (kirei) is learned as "pretty" or "beautiful."例：彼女はけっこうきれいだ。
(Eg: She's rather pretty.)
While it certainly can be used this way, don't be pigeonholed into using this one function of the word! You may not see it used differently in textbooks often, but actually in real life it quite often is.
I was at the dentist yesterday, and he asked me if I had braces as a kid. I said no and he seemed a little surprised and said:
歯がきれいだね。
Though I have no doubt that there must be some dentists out there who can really appreciate the true beauty of teeth, he wasn't telling me that my teeth were really pretty. In this case, きれい meant something closer to "well-ordered" or "neat," though I think it's a little difficult to translate exactly into English. He was telling me I had nice (straight) teeth.
Finally, きれい is also the opposite of 汚い (きたない；kitanai; dirty or messy). So it can be used to mean "clean" or "tidy."
Last week the teachers were cleaning their desks in preparation for moving seats or changing schools, and one nearby guy with a messy desk exclaimed at one point:
「あっ、きれいなってきた！」
"Look, it's finally starting to look tidy!"
*Update: Tokyo Five adds an interesting tidbit in the comment section -
When speaking to small children, mothers will often use the baby-talk and say "Kirei-kirei!♪" ("Clean up" or "Wash up").
You can also verbatize きれい: きれいにする - to clean, clear, tidy up.
例: お母さんは来週に来日するからアパートきれいにしなきゃなぁ。
Eg: My mom's coming to visit Japan next week, so I guess I need to clean my apartment.
