Rootkit is an application (or set of applications), that hides its presence or presence of another application (virus, spyware, etc.) on the computer, using some of the lower layers of the operating system (API function redirection, using of undocumented OS functions, etc.), which makes them almost undetectable by common anti-malware software.
Please note that rootkits can be either legitimate or malicious. Correct rootkits may be installed as a part of legitimate application. The list of some well-known rootkits can be found in the Detection of known legitimate rootkits. Because of that it is necessary to pay close attention to the Anti-Rootkit results.
Rootkit can get to a computer using various ways. The most common way is through some trojan horse or some suspicious email attachment. Also surfing the web may result in installation of a rootkit, for example when "special" plugin (pretending to be legitimate) is needed to correctly view some webpage, to launch some file, etc.