Well, usually we use извини. Прости is used pretty rarely, only in serious cases.
Извини — "Oh, I accidentally pushed you, sorry" and "Excuse me, where is the toilet?"
Прости — "Dear, I haven't been faithful to you, I accidentally killed your cat, burned your house and crushed your car. I'm SO sorry!" (and it is NEVER used like "excuse me"). Прости literally means "forgive me". And if the person forgives you, he/she says "Я тебя прощаю" or "я прощаю тебя" = "I forgive you".
You can say "извини" only to person with whom you are "на ты" (your parents, your friends, your spouse, your children, etc). That's familiar form.
"Извините" is the respectful form. If you barely know somebody and/or the person is older than teenager, you should call him/her/them "вы" and you should use only "извините" instead of "извини".
Also you can use "извините" for situation when you are guilty and you say "I'm sorry" to more than one person. For example, you and your friends have decided to go to the cinema together, but you were late and your friends had to wait for you. So you say to your friends: "Извините, что заставил (or заставила if you are female) вас ждать" — "I'm sorry to keep you waiting".
P.S. The same case with прости и простите. Прости is familiar, простите is respectful, official, appropriate for any case form.