Computer viruses
A virus is software that is installed without the users knowledge. The main thing that makes malware a virus is it's ability to self replicate. It will either copy itself within a computer system or between computer systems.
Some viruses are harmless but many can damage files or even fill up your hard drive with copies so it slows down your computer.
Trojans
The trojan horse famously provided a backdoor to a castle that was otherwise difficult to conquer. A trojan is a program that provides a back door for hackers to enter your computer. They might use this to upload further
malware, take control of your computer and use it to send phishing emails, to obtain personal details or pretty much any other reason they can think of.
Spyware
Spyware is software that is used to track your actions online. This can be both legal software designed to serve you more targeted content on a website or illegal software or even hardware like key loggers that can be used
to record every keystroke on a computer and thereby obtain personal information.