Verlet's method is a second order method for solving equations of motion that is more accurate than first order methods like Euler's method. It works by calculating the velocity at the midpoint of each time step using the acceleration from the beginning of the step. This time centering is what gives it second order accuracy. To calculate the final velocity and position, it uses the average of the velocities halfway between each step. Care must be taken to initialize the method by taking a half step back or forward in time to calculate the initial velocity.