A stiff differential equation is an equation that has some nasty behaviour in relation to numerical, discrete approximation schemes. There is no single precise mathematical definition of “stiff”, but several by several authors, including:
An equation is stiff if certain numerical approximations don’t work well, namely explicit numerical approximations.
An equation is stiff if the maximal step size that is needed for a stable numerical approximation varies greatly over the phase space of the equation.
Let’s say you move inside a closed room. Your movement is described by a stiff differential equation, and you use a discrete approximation to determine where you will be after the next timestep. You use an explicit scheme, which means you extrapolate your next position based on information about your current position only. It works for several timesteps, but suddenly your approximation of your next timestep says that your next position will be two meters behind the wall. Ouch! You should have used a smaller timestep, and you didn’t get a warning in time.
The concept of stiffness can be generalized to include stochastic differential equations.
Wikipedia, Wikipedia.
E. Hairer, G.Wanner: Solving Ordinary Differential Equations II, Stiff and Differential-A lgebraic Problems (ZMATH)