My favourite results are (1) the existence of NP-complete problems (Cook), (2) the Baker-Gil-Solovay theorem that whether P=NP holds relativized to on oracle depends on the oracle,
and (3) Faggin's characterization of NP in terms of second order logic.

I am not so much interested in the large number of proofs that show that a certain problem is NP-complete, but the fact that there is some problem that is NP-complete is remarkable
and important.  And Cook's SAT is actually natural.  (2) shows that several approaches will not work when one wants to settle P versus NP.  (3) gives a much more natural definition 
of the class NP.  Faggin's formulation (NP is the class of graph properties (of finite graphs) that can be expressed with a formula that has an n-ary second order existential quantifier in front, followed by a first order formula) indicates that NP vs co-NP is a very fundamental question as well (can second order existential quantification be replaced by second order universal quantification?).