Popular with single travellers for being hassle free and their relative cheapness, the airport bus goes from Suvarnabhumi on 4 seperate routes into the main tourist areas of central Bangkok. At 150B a person though, it's a 50% increase on the charge from the previous Don Muang airport and it's only likely to be better value than a taxi if you're travelling alone. In in a group of two or more a taxi is going to be quicker, more comfortable, more direct and either cheaper or rougly equivalent in price - so no contest really. The airport buses run from approximately 5am to midnight.
Bus routes and numbers:
Route AE-1 goes on the expressway from Suvarnabhumi to Silom road. You can connect to the Skytrain or subway from Silom.
Route AE-2 goes on the expressway to Banglamphu. This is the one to get on for Khao San road.
Route AE-3 goes to the Nana (soi 3-4) part of Sukhumvit road.
Route AE-4 goes from Suvarnabhumi to Hua Lamphong, Bangkok's main train station.