I've always assumed that the expansion ratio of an engine equaled the pressure inside the engine over the atmospheric pressure, but I'm not sure this is right, could someone check me on this?
If you're referring to nozzle expansion ratio, it's defined as the ratio of the nozzle exit area and throat area (the widest to the narrowest) and is often designated by greek letter epsilon.
$\epsilon = A_2 / A_t$
Where $A_2$ is the exit plane area and $A_t$ is the throat area. Usually nozzle expansion ratios are written in form 10:1 or just 10 (exit plane area being ten times the throat area). It is only a parameter of the nozzle design, not related to the chamber pressure.
You could also mean pressure ratio. When we talk about pressure ratio, it usually means the ratio between chamber pressure and exit plane pressure.
Where $p_1$ is the chamber pressure and $p_2$ exit plane pressure. Note, however, that the pressure on the exit plane is not necessarily equal to the pressure of the environment. The nozzle is designed to be at optimum expansion at certain altitude (so that athmospheric pressure equals the exit plane pressure), otherwise the nozzle is over- or underexpanded so the exit plane pressure would differ from that of the environment.
Rocket Propulsion Elements by G.P. Sutton explains these concepts very well.