Partial - Look for partial derivatives. None? Ordinary.
Linear - Look for exponents other that "1" on any derivatives? None? Linear.
Technically, in the second part of TKHunny's admirably brief explanation, you have to include derivatives of zero order (i.e., the function itself), in considering whether the DE is linear or not. In addition, you have to check whether there are nonlinear functions being taken of any derivatives (zero on up). The simple pendulum equation, before the often-done small-angle approximation, is nonlinear, even though it doesn't have any derivatives (zero order on up) to any power other than 1, unless you include the power series expansion of the sin function.