A root kit is a generic term for malware that actively tries to hide itself by modifying those parts of the operating system that you would ordinarily use to see the malware’s presence or operation.
For instance, an item of malware might be installed in a particular directory (folder) and be automatically started when the system boots by an entry in a system startup configuration file. Once you know what to look for it is easy to spot, either if the user looks, or a virus scanner goes looking. A rootkit modifies the operating system and kernel so that the system facilities that would revel the presence of the malware are blind to it. For instance the file system may have added to it a simple bit of code that blanks out the malware files from any file listing. The process viewer similarly is modified to not show the running malware process, and whenever you try to look inside the configuration files the system serves up a faked file that is not the one used a boot time. And so it goes. Network activity monitors are blinded, the file system will report the correct sizes of modified files, and in an ever escalating war, whenever a detection system adds a mechanism to detect the malware, a new new feature is added to the rootkit to either disable or disguise the result.
Within the taxonomy of malware, rootkits probably do count as proper viruses. They involve changes to the code of the target system (in the same manner that a biological virus adds itself to the DNA of an infected cell.) In some ways a rootkit is a little analogous to HIV, in that it actively attacks the defence systems of the host.
A related issue with rootkits, it the sophistication they have in hiding parts of themselves and of adding reinfection mechanisms. Because elements of the operating system have been infected, there is nothing left you can trust. If you delete files related to the malware, you can’t trust the operating system to have really deleted them, or you can’t trust the system not to still contain other code that will simply reinstall them. Worse, some modern rootkits hide components of themselves outside the normal parts of the filesystem. They can hide important components in unused parts of the disk, so they are easily available for installation, and a very nasty variant installs a re-installation program in the bootstrap sector of the disk - not a part of the the disk that is overwritten during a complete installation of the operating system, but one that is automatically executed every time you boot the computer.
Running detection software from a CD can be useful, since the CD is not writeable, and we hope has not been built from an infected source (not a complete given) we can reasonably assume it can’t be subverted, and will detect the malware. But, whether it is able to find every hiding hole the malware secretes itself in is another matter.
As they say in the classics, nuke it from space, is the only way to be sure.