A Baire space is a topological space that satisfies the conclusion of the Baire category theorem?.
It should not be confused with the space of irrational numbers (sometimes called ‘Baire space’ and coincidentally an example of a Baire space in our sense) or a Baire set? (a subset somewhat analogous to a measurable set but defined by a topological property).
A Baire space is a topological space such that the intersection of any countable family of dense open subspaces is also dense.
Any complete metric space (or rather its underlying topological space) is a Baire space.
Any locally compact Hausdorff space is a Baire space. In fact, any G-delta set of a locally compact Hausdorff space is a Baire space under the subspace topology.
Any open subspace of a Baire space is also a Baire space.
As mentioned above, the space of irrational numbers, or equivalently of infinite sequences of natural numbers, is also known as ‘Baire space’. It is a Baire space in the present sense (since it admits a complete metric), but not much should be made of the fact it has the same name.