Distributed Denial of Service (DDoS) is when an attacker sends large amounts of data to the target server. The goal is to stop the targeted server from communicating over the internet, thus 'denying access to services' to everyone that tries to connect to it. Most of these attacks simply send such large amounts of garbage data so that the internet connection of the targeted server gets jammed. Simply think of it as a traffic jam of sorts. For Tibia this means that usually that the outbound connection at the provider gets jammed, while the game servers are in fact running nicely. This is also the reason why characters die during such kicks. The server keeps on running normally, which means monsters are still attacking and killing.
Information taken from the official tibia site. See more here
See also Kick.