A denial of service (DoS) attack prevents a machine from carrying out its main duties -- for example, preventing a WebServer from serving up pages.
A distributed denial of service (DDoS) attack normally means many clients are sending repeated requests to a machine to saturate its bandwidth and/or processing capacity, preventing legitimate requests from getting through. Nowdays, the clients participating in such an attack are typically so-called "zombies", machines infected with a Virus or Worm containing remote control facilities, typically by joining a secret, password protected channel on some IRC server. They can then be instructed to when and where to attack from there. Such a group of zombies is known as a "bot net".