Importance of the Artin-Wedderburn theorem
Let $R$ be an arbitrary ring. The quotient $R/J(R)$, where $J(R)$ is the Jacobson radical, is the part of $R$ that acts nontrivially on the simple modules of $R$; said another way, it is the universal semiprimitive quotient of $R$. If $R$ is Artinian, then $R/J(R)$ is Artinian semiprimitive, which turns out to be equivalent to semisimple, so we can apply Artin-Wedderburn to $R/J(R)$ and learn something important about our original ring $R$ (namely everything we can learn from looking at simple modules).