Bayesian probability represents a level of certainty relating to a potential outcome or idea. This is in contrast to a frequentist probability that represents the frequency with which a particular outcome will occur over any number of trials.
An event with Bayesian probability of .6 (or 60%) should be interpreted as stating "With confidence 60%, this event contains the true outcome", whereas a frequentist interpretation would view it as stating "Over 100 trials, we should observe event X approximately 60 times."
The difference is more apparent when discussing ideas. A frequentist will not assign probability to an idea, either it is true or false and it cannot be true 6 times out of 10. A Bayesian however views ideas in a different light, the probability of an idea being true is equal to the number of possible worlds where it is true out of the total number of possible worlds.