Walk up to the person/thing you want to apologize to and say:
I'm sorry for ______, I hope you can forgive me.
Well that's what I do, anyway.
How can I say sorry?
Just do it, the sooner the better.
Just say it the way you said it on here. Tell them you really regret whatever happened and you're very sorry. If they're a decent person they will accept your apology.
______ [insert person(s) name(s)] I am sorry I ______ (offense). I hope you can forgive me.
Like everyone els said, yep.. Pretty straight forward.