If you have an analytic mind, you are good at using logic to figure things out. You don't let emotion sway you. Doctors are trained to think in this way, as are scientists and engineers.
Analytic derives from analysis, which means a thorough and detailed dissection of an object or an idea. Being analytic means that you think things through carefully and thoroughly. It may be more fun to fly by the seat of your pants and make all your decisions based on your gut instinct, but in the end, being analytic is safer.