Galina already gave you a few great examples of бывает's use but here are some more.
1) To visit a place, to be present at a place.
Он часто бывает у меня. (He often visits me).
Он бывает на рынке каждую неделю. (He goes to the market every week).
2) To occur.
Так не бывает! (This never could happen, it's impossible)
3) "Бывает!" - "It happens", said to console a person when something unpleasant has happened, one to console yourself if something happened to you.
- Я потеряла кошелёк. (I have lost my purse).
- Бывает.
Compare the saying "Бывают в жизни огорчения" that is used similarly.
4) "Бывает же!" expresses wonder, amazement. Similar to "Надо же!", "Ух ты!"
- Я слышал, вчера один человек прыгнул с пирса в море и случайно убил акулу. (I head yeasterday a man jumped from the pier into the sea and accidentially killed a shark.)
- Бывает же! (Oh boy!)