Why is there a rash around a tick bite?

Lyme. A rash around a tick bite, especially if its a target like red rash and you live in part of the country where lyme is common, is essentially diagnostic of infection with lyme disease. A relatively short course of oral antibiotics (doxycycline is usually best) will easily get rid of this.
Depends. A rash that occurs immediately after a tick bite is usually inflammation from the bite itself, or the tick burying in your skin, much like a splinter is irritating to the skin. A rash that occurs several days after a bite and especially with fever, chills, joint aches, or headache is much more concerning for lyme disease. If you have any doubts about a tick bite, see a doctor.