Let $X$ be an algebraic variety (in the modern approach an irreducible reduced scheme of finite type over a field, but we will work here with the usual maximal spectra to simplify the exposition). To a variety one can associate a function field $k(X)$ whose elements are rational functions on $X$ (they are regular functions on the big-Zariski open subvarieties of $X$). A partially defined map from a variety to another variety is rational if it is defined and regular on a Zariski open set. One can check easily that in fact rational maps compose. Rational maps which have an inverse on a Zariski open subset are called birational maps or birational isomorphisms (see there for a more precise definition).
Birational geometry considers properties of varieties which depend only on the birational class, i.e., they are equivalent when they have isomorphic function fields. In fact, one can define the appropriate category by starting with the category of varieties over a fixed field and then localizing at all birational equivalences.
Related articles: Mori program?, function field
János Kollár, Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics 134, Cambridge University Press 2010 (doi:10.1017/CBO9780511662560)
wikipedia: birational geometry, rational variety
Caucher Birkhar, Birationali geometry, online notes short version 35 pages, pdf; Lectures on birational geometry, 85 pages, arxiv/1210.2670
Bruno Kahn, R. Sujatha, Birational geometry and localisation of categories, arxiv/0805.3753v1
A recent new approach is in
Probing birational geometry via the derived category of coherent sheaves is explained in
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