Yep, check for a tight link, if that doesn't cure the problem, then you have a worn cassette/freewheel which has to be replaced. What happens is that as the chain, the weak link in the drivetrain (so to speak) wears, it wears the teeth on the cassette irregularly. When you put a new chain on, it doesn't mesh correctly with the worn teeth on the cogs and causes the skipping.

To avoid this problem in the future, check the chain for wear and replace well before the wear limit. I have put at least 4 new chains on the same cassette with no signs of skipping yet. Chains is cheap, cassettes, not so cheap.

Or, you can just put the old chain back on and keep riding it. It will last for a long, long time, but you run the risk of prematurely wearing out the chainrings if you go this route.