Back and again can be used with similar meanings, but there are some differences.

Back

Back is an adverb particle. When used with a verb back suggests a return to an earlier situation or position or a movement in the opposite direction. Again is not normally used with a verb in this way.

He had promised to give me my money back, but he didn’t. (NOT He had promised to give me my money again…)

I asked him to give me my watch back. (NOT I asked him to give me my watch again.)

The meat was stale so I took it back to the shop.

Again

When used with a verb, again suggests repetition.

I didn’t hear what you said. Can you say it again?

He asked the same question again and again.

Cases when back is not used

Back is not used when the verb itself suggests the idea of return to an earlier situation. In these cases again can be used to emphasize the idea of ‘return’.

Who opened the window? Could you close it again? (NOT Could you close it back?)

Call back

Call back can mean ‘return a phone call’ or ‘repeat a phone call’.

‘Jane is not in at the moment.’ ‘Could you ask her to call me back?’

‘Peter is busy. He doesn’t have time to talk now.’ ‘OK, I will call back later.’

In British English, ring back can be used with the same meaning.