My favorite theorem is probably a tie between $ e^{i\pi}+1=0 $, and the Pythagorean Theorem: $ a^2+b^2=c^2 $.
The former, because it incorporates all of the major constants: $ e $, $ \pi $, $ i $, $ 1 $, and $ 0 $. The latter because, though it is first taught very early in mathematics education, and remains useful for as long as you continue to study math (at least, it has over the course of my engineering education).
My favorite Theory is the Transonic Area Rule, which states that a body traveling through the transonic regime has minimum drag when the cross-sectional area of the body changes smoothly. Thus, a perfect aerodynamic shape according to this rule is known as a Sears-Haack Body. The first aircraft to be redesigned to accommodate this rule (accounting for the whole body) was the F-102 Delta Dagger, resulting in the F-106 Delta Dart. The resulting narrowing of the middle section of the aircraft came to be called a "wasp waist". A good picture of the F-106 on which the Area Ruling can be seen easily is here.
