For a given Lagrangian $L$, the $i$th generalized momentum is defined as $$p_i = \frac{\partial L}{\partial \dot{q_i}}$$ where $\dot{q_i}$ is the time derivative of the $i$th generalized coordinate (i.e. the $i$th generalized velocity).
I have also seen the above referred to as conjugate momenta, or even generalized conjugate momenta. What exactly is the difference between these terms? Do they all mean the same thing?