There are a number of network-based security attacks. They can involve a number of goals:
Programs which attempt to duplicate themselves to other systems are called worms (although often the term "virus" is used, somewhat inaccurately, instead; the difference is that a virus duplicates itself by adding itself to program executables, while worms typically run as stand-alone programs). The problem of worms has become significant now that so many Internet-connected hosts exist with vulnerabilities; although Microsoft Windows is the most common target of worms, any OS with an exploitable vulnerability can spread worms -- the first major worm, the Morris Internet Worm in 1988, targeted SunOS and BSD 4.2 hosts, and other worms since have targeted common OSes like Solaris and Linux.
Worms are now also combined with other security attack methods; besides installing back doors for attackers, some worms have also been used to spread software used in distributed denial-of-service (DDoS) attacks. Simple denial-of-service attacks originating from one host or network are easy to shut down -- just filter or turn off the host or network. DDoS causes small amounts of traffic to originate from a large number of hosts, making it harder to detect the individual origins but still resulting in crippling amounts of network traffic to the target.