Don Muang is the old airport.
Once Suvarnabhumi opened, Don Muang was going to be used for cargo and certain government business, but has been brought back into service for most domestic flights due to the **** job done on Suvarnabhumi (major cracks in runways among other things) which has meant that part of this new showpiece airport has been closed down while major repairs/rebuilding take place.

Don Muang will be used until the problems at the new airport are rectified (whenever that might be is anyone's guess).
You don't say where you're flying from but you'll probably be arriving at Suvarnabhumi as Don Muang is used for domestic flights (I don't think it's used for ANY international flights).

Depending on your time of arrival, you could be in downtown Bangkok in around 40 minutes (could be a fair bit longer if you arrive late aftenoon onwards).