Malware is a blanket term that refers to all the horrible software that assholes make to ruin your computer, such as viruses, trojans, worms and spyware
Malware is computer AIDS. It can range in horribleness from programs that pop up porno ads all the time to ones that email your credit card numbers to Latvia before erasing your hard drive. Malware has existed since the pre-internet dark ages, but it really became a problem after the internet became popular. The major types of Malware are viruses, trojans, worms and shitware.
This is the one everyone and their mom has heard about. Computers usually come with a type of antivirus software pre-installed on them nowadays (often one like Norton which is so badly made it actually kills performance worse than what it protects against).
Like the Greek Trojan Horse, a Trojan is a program that pretends to be a legit program but actually exists to run evil software which compromises the security of your computer.
Worms are malware whose primary vector for infection is the Internet. Often these worms are used to form botnets, which are large, interconnected networks of machines that have been transformed into "zombies" that are used to do the bidding of the bad guy who created the worm. Often these botnets are used for illegal activities such as spam email and distributed denial of sercie attacks.
Is often shovelware applications that install fun little features like porn pop ups and web browsing trackers that form the actual business model of the software. An early exponent of Spyware was Kazaa, a popular p2p software app that was often used to pirate commercial software, which usually contained other malware. Kazaa alone kept hundreds of mom & pop computer repair shops in business, in addition to probably being the main reason computer companies started moving their tech support lines over to India.
Types of Shitware:
Originally, the major malware development region was Eastern Europe, Bulgaria in particular, but now an increasing amount is coming from Asia as well.