Computer Viruses: These attach themselves to a programme or file in your computer and are designed to spread from one computer to another, leaving infections as they travel. Some viruses might not cause too much damage, while others may erase everything on your hard disk. Viruses are usually spread by attachments in emails or through IM, so you should NEVER open an email attachment if you do not know who the email is from. Viruses can also be spread via downloads from the internet or hidden in software or other files you might download.
Trojans: A Trojan Horse may initially appear as a useful piece of software being offered to you to run on your computer. People who receive Trojans will usually open them because they appear to be a legal piece of software from a reputable source. Once installed or run on your computer however, a Trojan will do damage. Some may do silly things like changing your desktop, others can cause more serious damage, like destroying information on your system or deleting files.
Spyware: Spyware is software which gathers your personal information by recording your activity while you are actually using your computer. There are a number of ways in which spyware can get onto your computer, sometimes a spyware piece of software will be installed on your computer while you are installing other software such as a music or video file sharing programme. Spyware is similar to Trojans since a user may be unaware that they are installing it while installing another programme. A common way to become a victim of spyware is to download certain peer to peer file swapping products.