Shikikin (敷金) is like a Western-style deposit: You pay money up-front, usually calculated based on the monthly rent of the apartment. "No deposit" apartments are becoming more popular, but 1 or 2 months' worth of shikikin is still standard as a downpayment, and for prime real estate it is possible an agent will ask for 6 months' of rent or more as shikikin. In theory, you get this money back when you move out of the apartment, minus the cost of damages incurred during your stay. However, stingy landlords are very good at finding damage where none exists. It is safer to assume you will not be getting your shikikin back at the end of your stay; if any money does make it back to you, you'll then be pleasantly surprised.
Shikikin is also sometimes called 保証金 (hoshoukin), which is closer to the Western term "deposit."
Reikin (礼金) is sometimes called "key money," but this term can be misleading. Reikin is, ultimately, just a gift to the agent for introducing you to and setting you up in the apartment. It usually equates to about 1 or 2 months' rent. But, again, it's becoming popular for agents to advertise "No reikin" properties in order to draw in more customers. Reikin is never refundable.
On top of shikikin and reikin, it is likely you'll also be asked to pay chuukai tesuuryou (仲介手数料), fire insurance (火災保険), and guarantor insurance (保証会社保険料), which we've explained details of in another article.