What causes an engine to backfire?

Answer 1

missfiring cylinder possibly due to fouled spark plug, ignition timing incorrect (leads swapped around). The reason for the backfire is unburnt fuel accumulates in the very hot exhaust system and ignites there.

Or...

If the engine is running and did not previously back fire, and now backfires when the throttle is rolled off (on the over-run) then the most likely cause is an air leak either in the inlet manifold, between the carburettor and the head, or in the exhaust system, often where the pipe joins to the engine, or where the silencer joins the pipe.

These backfires are due to a LEAN mixture failing to ignite properly on every cycle and so allowing unburned fuel to build up until it finally goes 'bang'

if it backfires but won't run at all, then it could be several things from the camshaft being 180 degrees out to the spark plug leads (on a twin or multi-cyclinder) being on the wrong plugs.
backfires occur when not all the fuel is ignited in the engine and instead ignites in the exhaust.