Though it may seem strange to hear, there is actually a hot philosophical debate about what probability really is. Though there are others, the two primary camps into which virtually all mathematicians fall are the frequentist camp and the Bayesian camp.
The frequentist interpretation describes probability as the relative likelihood of observing an outcome in an experiment when you repeat the experiment multiple times. Flipping a coin is a perfect example; the probability of heads converges to 50% as the number of times it is flipped goes to infinity.
The frequentist interpretation of probability is inherently objective; there is a true probability out there in the world, which we are trying to estimate.