ordinary differential equation
/ ODE (oh-dee-ee) /
Suppose the thing you are tracking depends on just ONE varying quantity — usually time, sometimes position along a line. The temperature of a single cooling object depends only on time. The amount of medicine in the bloodstream depends only on time. The height of a hanging chain depends only on position along it. When there is a single independent variable like this, the differential equation that governs it is called an ordinary differential equation.
Precisely, an ordinary differential equation involves derivatives of the unknown function with respect to a SINGLE independent variable, so only ordinary derivatives like dy/dx or d^2y/dx^2 appear — never partial derivatives. The unknown y is a function of one variable, y(x) or y(t). An equation like y'' + 3 y' + 2 y = 0 is ordinary because everything is differentiated with respect to the same one variable. The word 'ordinary' is simply there to distinguish it from the partial case.
Ordinary differential equations are the bread and butter of an introductory course, and they describe an enormous range of real systems: anything evolving in time as a lumped whole — a population, a falling body, an oscillating spring, a charging capacitor. They are usually far more tractable than partial differential equations, which makes them the natural place to learn the whole craft of differential equations before tackling the harder, multi-variable world.
dy/dt = -0.1 y is an ODE: the unknown y(t) depends only on time t, and only the ordinary derivative dy/dt appears. Contrast with a heat equation, where temperature depends on both position and time and partial derivatives appear.
One independent variable, one kind of derivative — that is what makes it ordinary.
The unknown function may itself be vector-valued (a system of several quantities), but as long as everything is differentiated with respect to one variable it is still an ordinary differential equation, not a partial one.