Realism is a family of positions rather than one, and the word is nearly useless without saying what it is realism about. Each version claims that some class of things exists independently of what anyone thinks.

The main disputes

About the external world. That physical objects exist whether or not they are perceived. The opposing view is idealism, and Moore's reply was that the claim is more certain than any premise an argument against it could use.

About universals. That properties exist over and above the things having them. Plato placed them in a separate realm; Aristotle placed them in particulars. The opposition is nominalism.

About morality. That moral claims state facts which hold independently of anyone's attitudes. The opposition includes expressivism, which treats moral statements as expressing attitudes, and moral relativism.

About science. That well-confirmed theories are approximately true and their unobservable entities exist. Anti-realists hold that theories are instruments for prediction, and whether electrons exist is a question the theory does not require you to answer.

The two standard arguments

No miracles. Scientific realists argue that the success of science would be miraculous if its theories were not approximately true. Predicting something novel and precise is hard to explain by luck.

Pessimistic induction. Anti-realists reply that the history of science is full of successful theories whose central entities do not exist — phlogiston, caloric, the ether. If past success did not track truth, present success is weak evidence too. See scientific method and paradigm shift.

Why the label misleads

Someone can be a realist about the external world and an anti-realist about morals, or a scientific realist and a nominalist about universals. The positions are independent, and treating realism as a single temperament is the most common error made with the term.