Bayesianism is the broader philosophy inspired by Bayes' theorem. The core claim behind all varieties of Bayesianism is that probabilities are subjective degrees of belief -- often operationalized as willingness to bet.
See also: Bayes theorem, Bayesian probability, Radical Probabilism, Priors, Rational evidence, Probability theory, Decision theory, Lawful intelligence, Bayesian Conspiracy.
This stands in contrast to other interpretations of probability, which attempt greater objectivity. The frequentist interpretation of probability has a focus on repeatable experiments; probabilities are the limiting frequency of an event if you performed the experiment an infinite number of times.
Another contender is the propensity interpretation, which grounds probability in the propensity for things to happen. A perfectly balanced 6-sided die would have a 1/6 propensity to land on each side. A propensity theorist sees this as a basic fact about dice not derived from infinite sequences of experiments or subjective viewpoints.
Note how both of these alternative interpretations ground the meaning of probability in an external objective fact which cannot be directly accessed.
As a consequence of the subjective interpretation of probability theory, Bayesians are more inclined to apply Bayes' Theorem in practical statistical inference. The primary example of this is statistical hypothesis testing. Frequentists take the application of Bayes' Theorem to be inappropriate, because "the probability of a hypothesis" is meaningless: a hypothesis is either true or false; you cannot define a repeated experiment in which it is sometimes true and sometimes false, so you cannot assign it an intermediate probability.
Bayesians conceive rationality as a technical codeword used by cognitive scientists to mean "rational". Bayesian probability theory is the math of epistemic rationality, Bayesian decision theory is the math of instrumental rationality. Right up there with cognitive bias as an absolutely fundamental concept on Less Wrong.
The term "Bayesian" may also refer to an ideal rational agent implementing precise, perfect Bayesian probability theory and decision theory (see, for example, Aumann's agreement theorem).