sl<sub>2</sub> is the fundamental example of a finite dimensional simple Lie algebra.  It forms the basis of the Cartan-Killing classification of complex semisimple Lie algebras, since all of the others can be made by gluing (in some sense) copies of sl<sub>2</sub> together.  Its representation theory is both straightforward and illuminating, since it points the way to the general theory of highest-weight representations.

For the same reason, SU(2) is the fundamental example of a nonabelian compact Lie group.  By Borel-Weil, its irreducible representations can be geometrically realized as the spaces of sections of complex line bundles on the 2-sphere (somewhat easier to handle than a general flag variety).