Malware is a word used to describe anything from silly viruses to trojans and keyloggers, it generally refers to programs and scripts that are specifically made to harm users and their systems.
Malware has often been made to spread trough E-mail and P2P networks, online file sharing. Etc.
Malware is typically used by hackers to gain unauthorized access to users systems, and personal information such as their bank accounts.
Adware will also sometimes fit into the category of malware, as it will often be designed in ways, which makes it difficult to remove from systems once it has been installed. Some of the most hated adware will just re-install itself, rename or move itself around, when the user attempts to uninstall it from their system.