The above answers deal mostly with finite-dimensional systems. As for the (systems of) PDEs, you typically need the Lax pair or a zero curvature representation (see e.g. the Takhtajan--Faddeev book mentioned in the wikipedia entry you linked to for the definition of the latter) or something else like that. To the best of my knowledge, the complete understanding of what is an integrable system for the case of three (3D) or more independent variables is still missing. In particular, for the 3D case the overwhelming majority of examples are generalizations of the systems with two independent variables. These generalizations are constructed using the so-called central extension procedure (e.g. the KP equation is related to KdV in this way). As for the reading suggestions, in addition to the Takhtajan--Faddeev book cited above, you can look e.g. into a fairly recent book Introduction to classical integrable systems by Babelon, Bernard and Talon, and into the book Multi-Hamiltonian theory of dynamical systems by Maciej Blaszak which covers the central extension stuff in a pretty straightforward fashion. Both books have extensive bibliographies with further references to look into.
Now, as for classification and identification of (new) integrable systems of PDEs, at least in two independent variables, it turns out that the (infinitesimal higher) symmetries play an important role here. A recent collective monograph Integrability, edited by A.V. Mikhailov and published by Springer in 20082009, could be a good starting point in this direction. See also another recent book Algebraic theory of differential equations edited by MacCallum and Mikhailov and published by Cambridge University Press. For a general introduction to the subject of symmetries of (systems of) PDEs, I can recommend the book Applications of Lie groups to differential equations by Peter Olver.