These DDoS attacks bombard targeted web servers with so much dummy traffic that people trying to access a site for legitimate purposes are unable to do so. It's most analogous to a traffic jam on a highway with no one able to move.
The attacks were focused on a company called Spamhaus, which maintains a "domain name system" to connect a typed-in URL to the correct server hosting the appropriate content. With this company's services compromised, large portions of the web became less stable.
A Dutch web host called Cyberbunker states that it will host anything except child pornography or terrorism-related content.