What causes a weird pressure in my calf when I run? It feels tight.

Cmpt syn. You may have an exertional compartment syndrome I would suggest a thorogh orthopedic exam which needs to be done to avoid potential nerve damage.
Many. There are many causes of pressure and tightness in the calves when running. It could just be deconditioned, strained, or overused. If it occurs every time after a certain distance and improves within a few minutes after stopping, it could also be "exercise-induced compartment syndrome" which is swelling inside a limited space caused by exertion.