Briefly, a realist about x holds that x enjoys mind-independent existence, that is, x exists regardless of whether anyone thinks, hopes or fears that x exists.

It may sound odd to demand of minds and other things mental that they have mind independent existence, but the claim, for instance, that my mind is mind independent just means that I have a mind regardless of whether anyone thinks, hopes, or fears that I do.
As well, a realist insists on there being explanations of the empirical world (including minds) in terms of the real world. Thus, a complete theory of the mind should explain the existence and functioning of minds in terms of the reality lying behind their empirically testable properties. This expectation strongly contrasts with the strictly empiricist position of phenomenalists.
Pete Mandik & Chris Eliasmith