A computer program that appears to have a useful function, but also has a hidden and potentially malicious function that evades security mechanisms, sometimes by exploiting legitimate authorizations of a system entity that invokes the program.
Often the delivery comes by way of deceiving the receiver into opening a file that is from a trusted source, but, is in fact, intended to cause harm. This is known as a phishing attack.
Once the trojan is installed on a machine, similar to a virus, the results can vary.
Like viruses, there are some trojans that cause more harm than others. They vary from being mildly annoying, such as inserting pop-ups to your screen every hour, to more sinister and malicious in their intentions and actions such as deleting your entire hard drive. Either way, being infected by a trojan is definitely something that is best avoided.
On the plus side however, unlike viruses and worms, trojans do not reproduce by infecting other files, nor do they self-replicate.