In my spare time I sometimes like to visit this website (Theorem of the Day): http://www.theoremoftheday.org/
On this website, they give theorems (one for each day). It inspired me to ask this question. What is your favourite theorem?
Here are the rules:
For example, I could probably pick any of the major theorems in Mathematical Logic or Theory of Computation, but I would probably pick Hall's Marriage Theorem for its pragmatic use (in the spirit of the question, here's the description from Theorem of the Day: http://www.theoremoftheday.org/CombinatorialTheory/Marriage/TotDMarriage.pdf )---the cornerstone of Matching Theory that plays a major role in the development of many algorithms, especially for graphs. I like the constructive proof for this theorem using bipartite graphs (though it is worth noting that Tutte's Theorem for general graphs is a generalization of this result).
What's your favourite theorem?