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:
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:

Wǒ xià cì zài lái.
我 下 次 再 来。
I'll come again next time. 

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:

Wǒmen xiān chīfàn zài tán.
我们 先 吃饭 再 谈。
We'll eat first, and then we'll talk.

再 as "another" or "some more"
The English word "another" is often avoided altogether by using 再.

Examples:

Nǐ zài ɡěi wǒ yī zhānɡ zhǐ.
你 再 给 我 一 张 纸。
Give me another piece of paper.

再 as "continuous action"

Here 再 is similar to the English "for a while longer" or "keep [going/doing]".

Examples:

Zài zhǎo zhǎo, zhǎo bú dào wǒmen jiù huíqù.
再 找 找，找 不 到 我们 就 回去。
Look a little more. If we can't find it, we'll go home.