This is violin/fiddle played in the famous "gypsy" (Romani) style. You might try looking up "gypsy violin" or "gypsy fiddle" as a start. This style has been very influential over the years. For a more classical take on it, try Bela Bartok's Romanian Dances or Braham's Hungarian Dances. For a more modern spin, look up "gypsy punk."
From the comments (@user1103)
It's indeed a violin, played by session musician Mark Feldman. TMBG strayed from the lounge-lizard feel of the original and styled it after Ottoman music from the era of the fall of Constantinople: That distinctive style is attributed to a number of Mediterranean stringed instruments, including a Turkish instrument called a Kemence.