再 (zài) goes before a verb and talks about something that has not happened yet: doing it again, doing a bit more, or doing it later, after something else. You already know it from 再见, "see you again".
Subject + 再 + verb. 再 always comes before the verb, never at the end of the sentence as "again" does in English. If there is a word like 能 or 想, 再 goes after it and directly before the main verb.
Adding 一下 or doubling the verb (看看, 问问) makes the request sound lighter and more polite. 请再说一下 is the phrase to use when you didn't catch what someone said.
With food, drink and shopping, 再 means "some more" or "another". It is often followed by 一点儿 or a number and measure word. Hosts in China say 你再吃一点儿吧 all the time.
再 can also mean "then" or "only after that". The first action comes first in the sentence, often with 了, and 再 introduces the second one: 吃了饭再去 means "go after we've eaten".
On its own with a time word, 再 puts something off until later. 明天再说 ("let's talk about it tomorrow") is a polite way to stop a conversation or delay a decision.
不要再 + verb + 了 tells someone to stop doing something, or not to do it again. 不想再 + verb says you don't want to do it again. 再也不 is stronger: "never again".
In questions, 再 stays in front of the verb: 你能不能再说一下? Note the tone change: 不 becomes bú before a fourth tone, as in 不要 búyào.
再 is only for repeats that haven't happened yet: plans, requests and suggestions. For something that already happened again, Chinese uses a different word, 又 (yòu), which you will meet at HSK 3. So "He came again yesterday" is 他昨天又来了, not 他昨天再来了.
A quick test: if the English sentence is about the past ("came again", "was late again"), don't use 再.
Practise this grammar point on paper: 18 exercises using only HSK 1 words, with an answer key on a separate page. Free to print and copy for your class.
Every example on this page uses only words up to HSK 1. Paste your own sentences into the level checker to see which words, if any, are above your level.