Radial engines - especially older large-bore designs like you find in WWII era warbirds - can suffer from a phenomenon known as hydraulic lock.
Basically while the engine is off and cooling some oil from the crankcase seeps past the piston rings in the lower cylinders, and collects there. When you try to turn the engine over with this oil in the cylinder the bottom piston contacts the oil (which is an incompressible fluid), and can't complete its travel down into the cylinder.
The stresses in the engine build until something - usually the connecting rod for that bottom piston - fails, resulting in an expensive repair bill and a grounded warbird.
To avoid this aircraft that are going to sit for a long time usually have a spark plug removed from the bottom cylinder and a hose run to an oil jug to collect the seepage.
In addition to this (or in place of it for shorter-term storage) the ground crew usually rotates the propeller through by hand enough times to ensure the bottom cylinder has gone through a compression stroke.
If there is oil in the bottom cylinder the crew will feel the hydraulic lock as extra force required to turn the propeller. They can then stop and remove a spark plug to drain the oil from the cylinder before attempting an engine start.
A similar phenomenon exists with rotary engines as Simon described in his answer, and the engine is pulled through for the same reasons.