"Malware" is a generic term for programs that try to secretly install themselves on your computer. It is any software that is designed to infiltrate or damage a computer system without the owner's informed consent.
Viruses are software that replicate or makes copies of itself and attaches itself to other pieces of software. Viruses require a host for survival and reproduction, just like their biological namesakes. Viruses must insert their code into an application like Word or Excel, or a data file for such programs, particularly ones that have macros or a scripting language capability.
Worms are self-replicating programs that do not need a separate software host. Worms are generally "network-aware" propagating by seeking out other connected computers with unpatched software or inadequate defenses.
Trojan (Trojan horse) is a program that installs malicious software while under the guise of doing something else. Though not limited in their payload, Trojan horses are more notorious for installing backdoor programs. Trojan horse programs depend on actions by the intended victims: the victim must run the Trojan. Generally they don’t replicate themselves.
Rootkits are designed to take control of computer systems. They hide their presence on a system and cover up what is really going on on your system. They change your operating system so it lies to you – hiding files, processes, registry keys and communication session.
Spyware intercepts or takes partial control over a user's interaction with the computer, without the user's informed consent. Spyware programs can collect various types of personal information, but can also interfere with user control of the computer in other ways, such as installing additional software, redirecting Web browser activity, or diverting advertising revenue to a third party. One form of spyware is Keyloggers. These literally record your everything you type (like passwords) and then send the information to a remote computer.
Adware is any software package which automatically plays, displays, or downloads advertising material to a computer.