All mathematicians are used to thinking that certain theorems are deep, and we would probably all point to examples such as Dirichlet's theorem on primes in arithmetic progressions, the prime number theorem, and the Poincaré conjecture. I am planning to give a talk on the history of "depth" in mathematics, and for that reason I would like to have a longer list of examples and, if possible, some thoughts about what makes them deep.
Most examples, I expect, will be from after 1800, but I am also interested in examples before that date.
When it comes to the meaning of "depth," I am interested in both specific and general explanations. In specific cases, one might point to the introduction of unexpected methods, such as analysis in Dirichlet's theorem, or differential geometry in the Poincaré conjecture, which are not implicit in the statement of the theorem. In most cases, it is probably not provable that these methods are necessary (e.g. there are "elementary" proofs of Dirichlet's theorem), but in some cases it is provable, by general theorems of logic. Both types of explanation are welcome.
Update. I am a little surprised that nobody mentioned reverse mathematics, which seems to offer a precise sense in which certain theorems are "equally deep." For example, on pages 36--37 of Simpson's Subsystems of Second Order Arithmetic there is a list of 14 theorems, including the Brouwer fixed point theorem and Riemann integrability of continuous functions, which are equally deep in a precise sense. Admittedly, these are not the deepest theorems around, but they're not shallow either. Later in the book one finds other results of equal, but greater, depth. How do MO members view such results?