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