I found several very nice proofs which I enjoyed:
1.Brilliant proof of fundamental theorem of algebra by Gian Maria Dall'Ara http://mathoverflow.net/questions/10535/ways-to-prove-the-fundamental-theorem-of-algebra/10684#10684
2.Some proofs of quadratic reciprocity: http://mathoverflow.net/questions/1420/whats-the-best-proof-of-quadratic-reciprocity (I especially liked that one: http://mathoverflow.net/questions/1420/whats-the-best-proof-of-quadratic-reciprocity/1431#1431)
3.Proof that $\mathbb{R}^{2n+1}$ does NOT have a square root (quite elementary and beatiful) http://mathoverflow.net/questions/60375/is-r3-the-square-of-some-topological-space/60389#60389
4.Nullstellensatz using model theory http://mathoverflow.net/questions/9667/what-are-some-results-in-mathematics-that-have-snappy-proofs-using-model-theory/9693#9693
5.If in ring R every countably generated ideal is principal than R is a PID http://mathoverflow.net/questions/8042/do-there-exist-non-pids-in-which-every-countably-generated-ideal-is-principal/8067#8067
6.An infinite dimensional vector space have smaller dimension than it's dual. http://mathoverflow.net/questions/13322/slick-proof-a-vector-space-has-the-same-dimension-as-its-dual-if-and-only-if-it/13372#13372
7.Topological proof that Z is a Bezout domain. http://mathoverflow.net/questions/42512/awfully-sophisticated-proof-for-simple-facts/64039#640397.
