Classical probability is the method that mathematicians use to assign probabilities of events occurring, such as in casino gambling, coin flipping or rolls of dice. By using numerical methods, statisticians examine the possible outcomes of an event, and then assign probabilities of those individual outcomes occurring. Classical probability would be used to determine the chance of choosing the eight of clubs from a single deck of 52 cards, for example.
Relative frequency approaches examine the long-run relative frequency of an event occurring. This is sometimes explained as the "average over time." In this notion of probability, the idea of the long run is over many thousands of iterations approaching an infinite numbers of trials. As such, only estimates of relative frequency are usually available; the accuracy of the estimates increases as mathematicians examine more iterations. For example, suppose you were interested in the probability of having a heart attack by age 50. The more patients who are examined, the better the estimate of the event occurring. As you are never able to examine every possible patient over the course of time, however, you can only estimate the relative frequency.
Subjective probability techniques are used when you are not able to mathematically estimate the probability of an event occurring. This is frequently examined as measuring a belief in an event's occurrence. For example, you may want to predict the color of the next car that drives around the corner. You base your beliefs on observation: You've seen mostly blue cars in town and you determine that the next car will be blue. This estimation lacks objective evidence, but in some cases this is the best estimation that is available.
Bayesian methods in probability draw on Bayes' rule, which implies that people's beliefs in the probability of an event occurring should be a compromise between prior beliefs of the event occurring and any relevant new evidence available. For example, if you know that 100 scientific studies have disproved the existence of extrasensory perception, and you conduct a study that proves ESP exists, you should be skeptical of the study's conclusion. You have overwhelmingly strong prior beliefs, which can weigh more heavily than the data in the one experiment.