Piercings can get infected and form an abscess. Sometimes the abscess goes away on its own, other times, antibiotics may be necessary. It's a good idea to check with the piercing artist and a doctor.
I know that unless there is immense swelling, the piercing is left in while the abscess drains and heals. Gargling with salt water is beneficial. If things don't start looking better after a few days though, it's probably a good idea to get a prescription for oral antibiotics.
By the way, some people are allergic to the jewelry that's put in during a piercing. An allergic reaction can cause swelling and infection too.