再【zài】

再 before a verb means "again" for something that hasn't happened yet: a request, a plan, a suggestion. It often comes with 一次 or 一遍 "once".

再 or 又

又 is "again" for a repeat that has already happened. 再 is "again" for one that hasn't.

• × 他昨天再来了 → 他昨天又来了

After 想, 要, 能

再 goes after a modal verb, not before it. 我想再去 is "I want to go again"; 我再想去 doesn't say that.

• × 我再想去一次 → 我想再去一次

"More" and "a bit more"

With an amount or an adjective, 再 is "more": another five minutes, a bit cheaper.

不再 for "no longer"

不再 says something won't happen any more.

再 also means "then, after that" in 吃了饭再走 "eat first, then go", which belongs with 先…再.