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There is more than one theorem by this name. In group theory, this refers to the statement that if a finite group G has a subgroup H, the order of H divides the order of G. In number theory, it is the statement that if p is a Prime number and f(x) is a polynomial of degree n, then f(x) = 0 (mod p) has at most n integral solutions for 0 < x < p. There is also Lagrange's four-square theorem, which states that any positive integer can be expressed as the sum of four square numbers. Proofs can be found in Wikipedia.

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Q: What is Lagrange's Theorem?
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