These buses connect both airportd in Bangkok, Suvarnabhumi or Don Muang airports, to the city center and especially to Khao San Road or Silom. Note that to go to Khao San Road, there are already buses S1 from Suvarnabhumi and buses A4 from Don Muang. LimoBus Airport Express are a bit more comfortable and normally a bit faster (they announce 40 minutes to 1 hour to reach Khao San Road or Silom) but they are also 3 times more expensive.
Both bus services serve Khao San Road and BTS stations Silom, Ratchadamri, Ploenchit and also Pratunam aea ( more info on official website).
The Limo Buses leave from Gate 8, Level 1 at Suvarnabhumi Airport and Terminal 1, Gate 7 or Terminal 2, Gate 14 at Don Muang Airport. There are ticket counters at these gates where you can buy tickets.
There are two bus lines connecting Don Muang Airport to Bangkok city. They allow you to go to BTS Mo Chit station or to join Bangkok Northern Bus Terminal (Mochit) or Victory Monument in central Bangkok.
The Bangkok Skytrain is called BTS. The trains have air condition and let you go over the heavy road traffic in the capital but you can not go everywhere even if connected to the Chao Phraya Express baots, it can let you go to major temples in Bangkok.
Taking the bus to get around is clearly not the easiest mean of transportation to use for a foreign tourist in Bangkok neither the fastest one but it's the cheapest one. Bangkok bus network is wide and you can go almost anywhere but just need to be patient.
Only Thai city for the moment to have a metro (there is metro project in Phuket), Bangkok is the city in Thailand that offers more choices of public or individual transport (metro, bus, boats, tuktuks...). No, there are not just tuk-tuks and pink taxis to get around Bangkok !