Steady white smoke; warm engine; any air temperature. This is bad news. The engine's coolant is leaking into the combustion chamber and creating a steady flow of steam. Caused by a leaking head gasket or a crack in a cylinder head, most commonly in the exhaust port area. Usually requires serious surgery on the engine.
Puff of white smoke during cranking, before the engine starts; cold engine; gas or diesel. This smoke is unburned fuel vapor being pumped out the exhaust pipe. The white is tiny droplets of fuel. In a gas engine, the spark plugs aren't firing, or the fuel mixture is too rich to ignite. It may mean the choke plate is stuck closed, or a load of bad gasoline. In a diesel, this cranking smoke indicates defective glow plugs, but may also come from low compression caused by worn rings or valves.