A generalized coordinate is one of a set of independent parameters specifying the configuration of a mechanical system subject to its constraints. An angle locates a bead constrained to a fixed circular hoop. Expressing kinetic energy and potential energy in these coordinates gives the Lagrangian and its Euler-Lagrange equations.
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The variables of the Lagrangian, e.g. the angles of a double pendulum. From that example it is clear that these variables don't need to be simple things like cartesian coordinates or polar coordinates (although these tend to be the overwhelming majority of simple case encountered): any way to describe the system is perfectly valid.
In quantum field theory, those variables are actually fields.