ききます is one of the most basic verb which students learn at a very early stage but it has a very extensive meanings and there are many things to remember when you use it.
The first meaning most students learn would be “to listen” or “to hear”.
ラジオをききます。= I listen to a radio.
CDをききます。= I listen to a CD.
One thing you need to remember in this meaning is that you can’t say 先生をききます, although in English it is perfectly OK to say “I listen to my teacher.” I don’t know why but you just can’t say it like that in Japanese. Instead, you must say 先生のはなしをききます. In this meaning the most likely kanji for it is 聴きます.
ききます also has a meaning of “to follow” or “to obey.” So if you say, 先生のいうことをききます, it means “I obey my teacher.” The kanji for this is 聞きます.
Another meaning of ききます is to ask (a question).
先生にききます means “I (will) ask the teacher.” The kanji for this is 訊きます although this kanji is not among the 常用漢字（じょうようかんじ, kanji designated for everyday use）.
ききます can also be written as 利きます or 効きます. The former of these means “function well” and the latter means “be effective.”
犬は鼻がよく利きます（いぬははながよくききます）= Dogs have a sharp nose.
薬が効きました（くすりがききました）= The medicine was effective.
When I learned English words like “have”, “make” and “take”, I was astonished to learn there are so many different meanings. ききます is one of those in Japanese!
