configuration space
Imagine collapsing the entire arrangement of a complicated machine down to a single point in an abstract space, where each axis is one generalized coordinate. As the machine runs, that point traces out a curve. That abstract space is configuration space, and the system's whole history is one path through it.
Formally it is an n-dimensional space (a manifold in general) whose coordinates are q_1, ..., q_n; a single point specifies the configuration completely. The system evolves as a path q(t). This is not the same as phase space, which also carries the momenta and so has 2n dimensions. For a single pendulum the configuration space is a circle; for a double pendulum it is a torus (a circle for each angle).
The principle of least action is a statement about paths in configuration space: among all neighboring paths sharing the same two endpoints, the actual motion is the one that makes the action stationary. Constraints confine the accessible configurations to a lower-dimensional submanifold.
The configuration space of a double pendulum is a torus: each angle lives on its own circle, so together they parametrize a doughnut surface.
Two angular coordinates make a torus, not a flat plane.
A single point in configuration space carries only positions; you cannot read off a velocity or momentum from it, which is why dynamics ultimately lives in phase space.