Malware (malicious software): noun - software that is intended to damage or disable computers and computer systems
Virus - A computer virus is a piece of malware that is designed to cause damage to your machine. This can be deleting programs or files, replicating to other machines, stopping you from using the internet, etc.
Trojan Horse - A Trojan Horse is essentially a virus that is just designed to look like benign software. Common Trojan Horses are designed to look like anti-virus software, or a shopping coupon program, but it's actually causing havoc on your machine.
Adware - A piece of malware that produces annoying pop-ups. This pop-ups may ask you to download other piece of software, or enter personal information.
Spyware - A piece of software that secretly records all your personal information, and sends it back to the programs creators.
Rootkit - A piece of software that is designed to gain unauthorized access to your computer, and remain undetected. It will usually mask itself as another piece of software.
Ransomware - Malicious software the encrypts your personal files on your computer, and threatens to delete your data if you don't pay the creators. Payment is usually in the form of bitcoin. This form of payment is anonymous and almost impossible to trace.
Worm - Is malicious software that replicates itself to other computers on a network. It's goal is to infect as many computers as possible.
If interested in learning about other types of viruses, you can find more here: