A bayesian is someone who views probability as a level of certainty relating to a potential outcome or idea. This is in contrast to a frequentist who views probability as a representation of how frequently a particular outcome will occur over any number of trials.
An event with a probability of .6 (or 60%) will be viewed by a bayesian as stating "I am 60% confident event X will happen in this next trial", whereas a frequentist 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.