Why is the Dirac operator so important - in both physics and mathematics? Why is the Dirac operator considered so important - in both physics and (pure) mathematics?
 A: It should be obvious why the Dirac operator is important in physics because of fermions. In mathematics, one could mention the following incomplete list.


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*Atiyah-Singer Index Theorem for the (twisted) spin complex, see e.g., Nakahara, Geometry, Topology and Physics, 1990; or Berline, Getzler and Vergne, Heat Kernels and Dirac Operators, 2004.

*Connes's noncommutative differential geometry.

*Schrödinger–Lichnerowicz formula.

*Kostant's cubic Dirac operator.
For further information, see also nLab.
A: Because it describes spinors (i.e. behavior of fermions).  String Theory speaks about both bosons and fermions... when passing to a QFT, fermionic wavefunctions become spinor fields, and their dynamics require the Dirac operator.  And this is where all the mathematical importance stems from: QFT and String Theory (topological field theories) is a pure math theory.  In particular, I disagree completely with Noldorin's seemingly naive comment... Clifford algebras and spinors on manifolds is a huge field.
Sorry if this too brief of an explanation (for immediate purposes).
