The German army was the most notable user of motorcycle sidecar combinations. They were most commonly used for reconnaissance purposes. They had the advantage of being able to turn around in a normal road width for quick withdrawal and the sidecar could be used to mount a machine gun or a light mortar. They served very well in Western Europe where there was a well developed road network but their shortcoming were revealed in USSR where they had great difficulty dealing with the primitive infrastructure.

The other common use of motorcycles, without sidecars, was as dispatch riders/messengers. These were used by most countries to deliver orders and summon officers to conferences. In the Commonwealth armies of WW2 most officers were trained in motorcycle riding.

Eventually the reconnaissance function was mostly taken over by light cars such as the Kubelwagen, Schwimmwagen, jeep, etc. as well as armoured cars and 1/2 tracks.

It's also worthy of note that the German sidecar combinations had powered drive to both rear wheels.