What is Explicit, implicit, theta and Crank–Nicolson schemes?

With time step $\Delta\tau$, the theta scheme is $(I - \theta\Delta\tau L_h)U^{n+1} = (I + (1 - \theta)\Delta\tau L_h)U^n$. It is the explicit scheme (forward Euler) for $\theta = 0$, the implicit scheme (backward Euler) for $\theta = 1$ and the Crank–Nicolson scheme (Crank and Nicolson, 1947) for $\theta = \frac12$; its time error is $O(\Delta\tau)$ except at $\theta = \frac12$, where it is $O(\Delta\tau^2)$.