A very simple explanation follows...
A virus is a small piece of software or code designed to piggyback itself to a program in your computer. Every time that program is run, the virus also runs and has a chance to replicate or reproduce itself and then attach to another program in the computer. Usual symptoms are the program being used does not operate normally, and it operates more slowly.
These viruses migrate by means of e-mail messages and/or attachments. They spread by e-mailing themselves to every address found in the address book in the computer. Given the sheer volume of e-mail traffic on a global basis, this allows these viruses to spread very rapidly, and the volume created has forced e-mail server shutdowns on a number of occasions.
These are small pieces of software that makes use of computer networks and security holes found in them to replicate and spread. Most worms are written to detect and exploit a specific security hole or flaw. Once a computer on a network is discovered with the appropriate weakness, it gets attacked and infected by the worm. The worm then scans the network looking for another computer with the same hole and the process repeats. Now there are two computers for it to replicate from. The process continually repeats itself, but with the speed of today’s computers and networks, a network of say 50 computers and a properly engineered worm can easily infect all 50 computers in the network in under an hour. Perhaps the most famous worm of recent times was Code Red. In July of 2001 it replicated itself over 250,000 times in just nine hours.
Trojan (Trojan Horse)
Simply stated, a Trojan is a program. The program claims to do one thing, but when run, it does damage to the computer running it (for example, it may be designed to erase your hard drive). Fortunately, a straight Trojan Horse has no way of replicating itself.
Malware, short for malicious software, is software designed to infiltrate a computer system without the owner's informed consent. The expression is a general term used by computer professionals to mean a variety of forms of hostile, intrusive, or annoying software or program code. The term "computer virus" is sometimes used as a catch-all phrase to include all types of malware, including true viruses.
Software is considered to be malware based on the perceived intent of the creator rather than any particular features. Malware includes computer viruses, worms, trojan horses, most rootkits, spyware, dishonest adware, crimeware and other malicious and unwanted software.