the master equation
The master equation is the differential equation governing how the probability distribution of a continuous-time Markov chain evolves in time. In physics and chemistry it is the standard name; in probability it is the Kolmogorov forward equation. It is the continuous-time chain's analogue of one matrix multiplication per step in a discrete chain, written instead as a flow of probability among states, and it is the equation you actually solve (numerically or in closed form) to find time-dependent behavior.
Write p_t(j) = P(X_t = j) for the distribution at time t, started from some initial law. The forward (master) equation is dp_t(j)/dt = sum_{i} p_t(i) q_ij = sum_{i != j} p_t(i) q_ij - p_t(j) q_j. Read it as a balance of probability flux: the rate of change of being in j equals the inflow (probability currently in other states i times the rate q_ij into j) minus the outflow (probability in j times the total exit rate q_j). In matrix form, with p_t a row vector, this is dp_t/dt = p_t Q, equivalently the transition function obeys dP_t/dt = P_t Q (forward) and dP_t/dt = Q P_t (backward). The backward equation differentiates in the starting state and is the natural one for computing expectations of functions of the future: u_t(i) = E_i[f(X_t)] solves du_t/dt = Q u_t. Both are solved by P_t = e^(tQ) when the chain is non-explosive.
The master equation is the working tool for time-dependent CTMC questions: relaxation to equilibrium, transients in queues, chemical kinetics, gene-expression noise. Two honest cautions. First, on an infinite state space the forward and backward equations are NOT automatically equivalent and the minimal process is the unique solution only under non-explosion; if the chain explodes there are multiple honest 'solutions' depending on what is done at the explosion. Second, the master equation is exact for the probabilities but is an infinite (or huge) coupled system; the deterministic 'rate equations' of chemistry are only its mean-field limit and ignore fluctuations, which the master equation does keep.
For the two-state chain with Q-rows ( -lambda, lambda ) and ( mu, -mu ), the master equation for p_t = (p_t(0), p_t(1)) is dp_t(0)/dt = -lambda p_t(0) + mu p_t(1). With p_0(0)=1 it solves to p_t(0) = mu/(lambda+mu) + lambda/(lambda+mu) e^(-(lambda+mu)t).
The master equation as probability bookkeeping: inflow minus outflow for each state, giving relaxation to equilibrium at rate lambda+mu.
On an infinite state space the forward and backward equations are equivalent and have the unique solution P_t = e^(tQ) only when the chain is non-explosive; if it can explode there are several boundary behaviors and several solutions. The deterministic chemical rate equations are merely the mean-field limit of the master equation, not the same thing.