Light apologies (everyday)

3

For bumping into someone, interrupting, getting attention. Not a real apology, just politeness.

不好意思 bù hǎo yìsi excuse me / pardon (everyday "excuse me")
麻烦了 máfan le sorry to bother you (when asking for a favor)
打扰了 dǎrǎo le sorry for the disturbance (entering an office, interrupting)

Standard apologies

3

When you actually did something that warrants an apology.

对不起 duìbuqǐ I'm sorry (standard, sincere)
抱歉 bàoqiàn my apologies (slightly more formal than 对不起)
是我的错 shì wǒ de cuò it's my fault (explicit acknowledgment)

Strong or formal apologies

3
非常抱歉 fēicháng bàoqiàn sincerely sorry (formal, letters, customer comms)
请原谅 qǐng yuánliàng please forgive me (explicit request for forgiveness)
我错了 wǒ cuò le I was wrong (sincere admission)

Responding to an apology

3
没关系 méi guānxi it's okay (standard)
没事 méi shì it's nothing (casual, spoken)
别担心 bié dānxīn don't worry (reassuring)