A theory of emotion which implicates people’s personal interpretations of an event in determining their emotional reaction. The most  important part of this theory is the way we interpret the event (aka, was the event a positive or a negative occurrence?) as well as what we think caused the situation.

This website explains it very well: http://changingminds.org/explanations/theories/cognitive_appraisal_theories.htm for instance, here is a little diagram from that link:

Event ==> thinking ==> Simultaneous arousal and emotion

Example: Two professors react after their colleague is promoted; one feels happy for her friend, while the other feels resentful because he thinks he is more deserving of the promotion.