firstly if your chain is skipping you've left changing it WAY too late and will need to replace your chain and your cassette. The badly worn chain will have badly worn your cassette and a new chain will skip on it as well. You'll be lucky if your chainrings don't skip too, although try it and see.
Secondly you should buy this, a chain wear tool, it's like a $5 thing and it tells you when your chain needs replacing. If you replace chains on time, then your cassettes and chainrings (much more expensive) last a lot longer.
thirdly chains are basically all compatible for the number of cogs on your cassette, 7,8,9 etc.
7 and 8 speed chains are compatible
I would recommend KMC brand as they're easier to change.
You also need a chain breaker tool to cut it to size when installing. Look at Sheldon Brown for how to size a chain.
and overhaul your jockey wheels when you change it.
You can look at RJ the bike guy for tutorials on youtube about how to change a cassette.