I was wondering what is in the mosquito bites, which causes the swelling of the skin at the bite and also the itching. I know that the mosquito injects some stuff, but what is it?
When female mosquitos bite to suck blood, they need to overcome the body reaction against it. So when they found a bood vessel, they inject their saliva into the wound. This saliva contains substances which affect blood clotting and platelet aggregation (so the blood will stay liquid which is essential for sucking) and a vasodilatory substance (which widens the blood vessel and prevents contraction of it). The exact composition depends on the exact mosquito species.
These "foreign" substances cause a local immune reaction which results in the release of histamine in the skin. The itching and also the red skin and the swelling are a result of the histamine release. See the Wikipedia article on mosquitos for more information.
If you are interested in a deeper look on the composition of the mosquito saliva have a look at these two articles and their references (if there is help needed in getting the articles, let me know):
Since the substances affect the blood coagulation cascade, there is some research going on if and how this can be used clinically. See this article for an overview or have a look at the original publication here ("Unique thrombin inhibition mechanism by anophelin, an anticoagulant from the malaria vector")