A mosquito does not actually bite you, of course. It sucks your blood.

To help enable effective blood sucking, it first injects anti-coagulant saliva to stop the blood from clotting or forming a scab while it feeds. When the mosquito goes away, its saliva stays in the pocket under your skin.

Now comes the itchy part. Your body releases histamine to fight off a foreign substance. It is the same as an allergic reaction. The histamine causes swelling around the area and as a side effect, it itches.