Negligence is the legal term for any careless behavior that causes, or contributes to, an accident. For example, a person is negligent if he neglected to stop at a stop sign and, as a result, hit your car as you were coming through the intersection.
A person can be considered negligent whenever he or she had a duty to act carefully and failed to do so. (Generally, we all have an obligation to act with ordinary and reasonable care in any given situation -- that is, in a manner that will not foreseeably injure those around us.) For example, a person who drove at night wearing sunglasses would be negligent, because any reasonable driver would know that doing so would increase the chances of causing a traffic accident.
For most types of accidents, a person must be found negligent in order to be held legally responsible for another person's injuries. If a person behaves negligently and that behavior causes you harm, you can most likely recover compensation for your injuries.
For more about the basic rules used to determine carelessness, see the article Proving Fault in Personal Injury Accidents: General Rules.