When someone calls you quick-tempered, they mean that you get angry really easily. If you find yourself yelling at people several times a day, you may be quick-tempered — and you may want to take up meditation.
Quick-tempered people are irritable and a little bit unpredictable. You could also describe them as "short-tempered" or "hot-tempered." Some people can laugh off things that irritate or aggravate them, or think them over thoughtfully, or just ignore them. Everyone else is quick-tempered. This adjective uses the noun sense of temper, which can mean both "state of mind" and specifically "angry state of mind."