To begin with, it is only the female mosquitoes that bite.
What is known as a mosquito bite is actually a jab by the mosquito using a long-tube-with-a-sharp-end, proboscis (pronounced pro-bo-sis or pro-bos-kis), which she uses to draw blood from our blood vessels.
The moment a mosquito bites us, our body sensing an injury triggers the hemostatic response, i.e. it activates the system to clot the blood so there is no loss of it.
This makes the job of the mosquito difficult, so she secretes inhibitors (an anticoagulant and proteins) that prevent the blood from clotting.
As with any foreign invasion, when the brain observes chemicals injected by the mosquito, it activates the immune system to stop or control it.
One of the things the immune system does to stop the spread of a foreign chemical is to send its own chemical called histamine to the site.
Histamine causes the blood vessels surrounding the injured area (area of the bite) to become wider so that fluid and cells of the immune system, such as leukocytes (white blood cells) and blood plasma proteins can reach the site faster to fight the infection.
The fluid, filled with white blood cells sent over to the site explains why we get the bump on the skin.
While histamine is good to speed up the work of the immune machinery, we are mildly allergic to it.
It activates the nerves involved in itching, which is why mosquito bites make us scratch.
The itchy sensation is not yet fully understood by science but it has been found to be a subclass of the pain signal (it is pain but not pain and is confusing to the brain as well).
When we scratch, the action of our nails sends a low-level pain signal to the brain that overrides the itching sensation.
But now at least the brain can understand the signal and this understanding creates an illusion of relief.
