While 又 (yòu) is used for "again" in the past, 再 (zài) is used for "again" in the future. That is, 再 is used when something has happened once and it will happen again in Chinese grammar. It is Expressing time and date.
再 as "again"
The basic structure for using the future "again" is:
subject+再+verb+object
In fact, this structure is present in one of the most common Chinese phrases: "再见!" In this case, it literally means "See you again!"
Examples:
我 下 次 再 来。
再 as "and then"
A usage that feels sort of related but deserves separate mention is the use of 再 to mean "and then." It is often preceded by 先 when giving clear sequential orders ("first do this... then do this..."), but it can also appear by itself to mean "and then."
The example:
我们 先 吃饭 再 谈。
再 as "another" or "some more"
The English word "another" is often avoided altogether by using 再.
Examples:
你 再 给 我 一 张 纸。
再 as "continuous action"
Here 再 is similar to the English "for a while longer" or "keep [going/doing]".
Examples:
再 找 找,找 不 到 我们 就 回去。