Example sentences
zài再jiàn见！
Goodbye!
zài再jiàn见，míng明tiān天jiàn见。
Bye, see you tomorrow.
About this word
再 means again and 见 is to see or meet, so 再见 is literally "meet again". English does the same thing with "see you", and so do plenty of other languages; the Chinese version simply froze into a set phrase. 再见 is neutral and slightly formal, the safe choice in a shop, a classroom, or with someone you do not know well. Among friends a concrete version sounds warmer: 明天见 (see you tomorrow), 一会儿见 (see you in a bit).
The story behind the character
再
再 shows a fish with two horizontal lines, or something bound twice, and nobody has settled which. Xu Shen gave it a clean reading: 再, 一举而二也, one motion that yields two. That is how 再 came to mean again, a second time. Old Chinese kept 二 and 再 apart, since 二 was two things side by side.
见
目 (eye) over 人 (person). A person with a wide-open eye. That's 见. Xu Shen wrote 见, 视也: "to see." The simplest compound there is, no trick to it.