What considerations are important when purchasing a new chain?
The two most common sizes are 3/32" and 1/8"  Generally you would use 3/32" for bicycles which have dérailleur gears (road/mountain bikes).  As for length I believe they are all a stock length and you'll be very unlucky if you find it is too short.  You will need to specify how many speeds your freewheel or cassette is (6/7/8/9/10 speed).

You will obviously find lots of variety in quality/weight/material/price etc when you know which type of chain to look at.