It's kind of surprising how many branches of mathematics are structured around some kind of single fundamental theorem. Maybe in some cases, there are more than one theorem that compete for the title, but often there is a clear winner. Here are some fundamental theorems:
Arithmetic - Every positive integers is the product of a unique multiset of prime factors.
Algebra - Every complex polynomial of degree n is the product of n affine factors.
Calculus - Riemann Integration and differentiation are inverses.
Analysis - Lebesgue integration is the same as Riemann integration.
Game Theory - In a matrix game, both players have a mixed strategy which stochastically gurantees a common average outcome, at worst.
Statistics - The normalized limiting distribution of independent, bounded variance random variables is Gaussian.
Data Compression - A random variable can be decodably compressed into an average number of bits equal to its binary entropy, but no smaller.
Channel Coding - A channel can be used to reliably transmit a number of bits equal to its binary capacity, but no more.
Geometry - Any polygon that has n vertices also has n sides. Just kidding, there is no fundamental theorem of geometry, and don't tell me it's the Pythagorean theorem.
Galois Theory - For any n, if n is the power of a prime then there is a unique field of size n, otherwise no field of size n exists.