I forgive you would be for something sincere and serious, like a hurtful insult or even serious crimes.
For something formal or written, you can say I accept your apology.
A very common phrase, as @CookieMonster mentions, is It's OK or It's all right.
Sometimes, people apologize out of politeness, just in case the other person might be offended. In such a situation, people often respond with No worries or No problem. For example, if you texted your friend that you were running ten minutes late to meet her, hopefully she would say, "No worries!"
Another very common phrase is No need to apologize.
There are other more colorful expressions, such as:
Don't worry about it.
Forget about it!
Don't even stress!
Don't even trip!
If you don't forgive someone, you can usually add "not" or "don't" to the phrases above:
I don't forgive you.
It's not OK.
That's not good enough!
Talk to the hand!