What is wrong with my big toe if it is swollen?

Many possibilities: Could be a fracture, strain, sprain, infection, gout, many different systemic diseases, etc. See a podiatrist for a physical exam and possibly x-ray.

Many possible causes: Some local causes of swelling can include gout, arthritis, infection, lymphatic obstruction, ingrown nail, and trauma (fracture, sprain, tendonitis, tight shoes) have it evaluated by a professional and get the appropriate treatment.