A virus is a program which spreads itself by modifying other executable files, which are later unknowingly passed along to other users.
Viruses often contain a malicious payload, which may make the computer behave erratically or delete files.
A trojan horse, or trojan for short, is a malicious program disguised as a desireable one. The term comes from Greek mythology, where a wooden horse secretly containing soldiers was left as a "gift" to the citizens of Troy. After the horse was brought inside, the soldiers came out and attacked the city.