What is the rabbi called who performs circumcision, and what is the name of the ceremony itself?

Circumcision is known as Brit Milah or Bris Mila (there are various spellings). It is performed by a mohel. Many mohels are rabbis, but I don't think a mohel actually has to be a rabbi.

Brit Milah (accent on the 'lah') is the Sephardic pronunciation; Bris Mila (accent on the 'Mi') is Ashkenazic pronunciation.
Interestingly, up until recently male members of the British royal family were circumcised by a Jewish mohel.

'Mohel' is spelling everywhere for the person using the knife, though Ashkenazim tend to pronounce the word 'moyel'. He can be anyone who has mastered the procedure and ritual after study and apprenticeship with an established mohel. The sandek is the person honoured with holding the baby. The Mohel should get the fee. Our mohel just took his expenses and left the rest.

http://www.britpro.com/default.asp?p=procedure