Realists tend to believe that whatever we believe now is only an approximation of reality and that every new observation brings us closer to understanding reality. Realism is contrasted with anti-realism.
The oldest use of the term comes from medieval interpretations and adaptations of Greek philosophy. In this medieval scholastic philosophy, however, "realism" meant something different -- indeed, in some ways almost opposite -- from what it means today. In medieval philosophy, realism is contrasted with "conceptualism" and "nominalism". The opposition of realism and nominalism developed out of debates over the problem of universals. Universals are terms or properties that can be applied to many things, rather than denoting a single specific individual--for example, red, beauty, five, or dog, as opposed to "Socrates" or "Athens". Realism in this context holds that universals really exist, independently and somehow prior to the world; it is associated with Plato. Conceptualism holds that they exist, but only insofar as they are instantiated in specific things; they do not exist separately. Nominalism holds that universals do not "exist" at all; they are no more than words we use to describe specific objects, they do not name anything. This particular dispute over realism is largely moot in contemporary philosophy, and has been for centuries.
Increasingly these last disputes, too, are rejected as misleading, and some philosophers prefer to call the kind of realism espoused there "metaphysical realism," and eschew the whole debate in favour of simple "naturalism" or "natural realism", which is not so much a theory as the position that these debates are ill-conceived if not incoherent, and that there is no more to deciding what is really real than simply taking our words at face value.
Some realist philosophers prefer deflationary theories of truth to more traditional correspondence accounts.
Mathematical realism, like realism in general, holds that mathematical entities exist independently of the human mind. Thus humans do not invent mathematics, but rather discover it, and any other intelligent beings in the universe would presumably do the same. In this point of view, there is really one sort of mathematics that can be discovered: Triangles, for example, are real entities, not the creations of the human mind.
Many working mathematicians have been mathematical realists; they see themselves as discoverers of naturally occurring objects. Examples include Paul Erdős and Kurt Gödel. Gödel believed in an objective mathematical reality that could be perceived in a manner analogous to sense perception. Certain principles (e.g., for any two objects, there is a collection of objects consisting of precisely those two objects) could be directly seen to be true, but some conjectures, like the continuum hypothesis, might prove undecidable just on the basis of such principles. Gödel suggested that quasi-empirical methodology could be used to provide sufficient evidence to be able to reasonably assume such a conjecture.
Within realism, there are distinctions depending on what sort of existence one takes mathematical entities to have, and how we know about them.