canonical coordinates

Canonical coordinates are the 'correctly paired' variables of Hamiltonian mechanics -- a set of positions q and momenta p matched up so that the equations of motion take the clean canonical form. The word canonical here means 'in accordance with the rule': these are the coordinates in which q_dot = partial H / partial p and p_dot = - partial H / partial q hold, and in which the deeper structure of the theory is manifest. Not every pair of variables you might invent qualifies; being canonical is a property to be checked.

The defining test is the fundamental Poisson brackets: coordinates (q_i, p_j) are canonical if {q_i, q_j} = 0, {p_i, p_j} = 0, and {q_i, p_j} = delta_ij (the Kronecker delta, 1 if i = j and 0 otherwise). Equivalently, they are the pairs conjugate under the symplectic form. Each q_i has exactly one partner p_i, its conjugate momentum, defined originally as p_i = partial L / partial q_dot_i. Crucially, 'momentum' here need not be mass times velocity: the momentum conjugate to an angle is an angular momentum, the momentum conjugate to a field is another field, and for a charged particle it is p = m v + q A, including the vector potential.

Canonical coordinates matter because the freedom to change them -- via canonical transformations that preserve the fundamental brackets -- is the great power of the Hamiltonian formalism. You can trade an awkward set for one in which the Hamiltonian becomes trivial (this is the goal of Hamilton-Jacobi theory and of action-angle variables). The whole machinery of Poisson brackets, Liouville's theorem, and canonical quantization is phrased in terms of canonically conjugate pairs.