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.
The time derivative of a generalized coordinate. It is an independent velocity argument of a mechanical Lagrangian, but need not have the units or direction of a Cartesian velocity: for an angular coordinate it is an angular velocity. A regular Legendre transform replaces these variables with the corresponding canonical momenta.
Articles by others on the same topic
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.