An ordinary differential equation (frequently called an "ODE," "diff eq," or "diffy Q") is an equality involving a function and its derivatives. An ODE of order n is an equation of the form
F(x, y, y', y'', ...) = 0 (1)
where y is a function of x, y' = dy/dx is the first derivative with respect to x. Order refers to the highest differential in the equality.
A partial differential equation (PDE) is an equation involving functions and their partial derivatives;
F(u(x, y), u(x, y)', u(x, y)'', ...) = 0
an example of which is the wave equation