The millenium seemed to spur a lot of people to compile "Top 100" or "Best 100" lists of many things, including movies (by the American Film Institute) and books (by the Modern Library). Mathematicians were not immune, and at a mathematics conference in July, 1999, Paul and Jack Abad presented their list of "The Hundred Greatest Theorems." Their ranking is based on the following criteria: "the place the theorem holds in the literature, the quality of the proof, and the unexpectedness of the result."
The list is of course as arbitrary as the movie and book list, but the theorems here are all certainly worthy results. I hope to over time include links to the proofs of them all; for now, you'll have to content yourself with the list itself and the biographies of the principals.
1 
Pythagoras and his school 
500 B.C. 

2 
1799 

3 
1867 

4 
Pythagoras and his school 
500 B.C. 

5 
Jacques Hadamard and CharlesJean de la Vallee Poussin (separately) 
1896 

6 
Godel’s Incompleteness Theorem 
1931 

7 
1801 

8 
The Impossibility of Trisecting the Angle and Doubling the Cube 
1837 

9 
225 B.C. 

10 
Euler’s Generalization of Fermat’s Little Theorem 
1760 (1640) 

11 
300 B.C. 

12 
Karl Frederich Gauss, Janos Bolyai, Nikolai Lobachevsky, G.F. Bernhard Riemann collectively 
18701880 

13 
1751 

14 
Euler’s Summation of 1 + (1/2)^2 + (1/3)^2 + …. 
1734 

15 
1686 

16 
Insolvability of General Higher Degree Equations 
1824 

17 
1730 

18 
Liouville’s Theorem and the Construction of Trancendental Numbers 
1844 

19 
1770 

20 
All Primes Equal the Sum of Two Squares 
? 
? 
21 
1828 

22 
The NonDenumerability of the Continuum 
1874 

23 
Formula for Pythagorean Triples 
300 B.C. 

24 
The Undecidability of the Coninuum Hypothesis 
1963 

25 
SchroederBernstein Theorem 
? 
? 
26 
Leibnitz’s Series for Pi 
1674 

27 
Sum of the Angles of a Triangle 
300 B.C. 

28 
Pascal’s Hexagon Theorem 
1640 

29 
Feuerbach’s Theorem 
1822 

30 
The Ballot Problem 
1887 

31 
Ramsey’s Theorem 
1930 

32 
The Four Color Problem 
Kenneth Appel and Wolfgang Haken 
1976 
33 
Fermat’s Last Theorem 
1993 

34 
Divergence of the Harmonic Series 
1350 

35 
Taylor’s Theorem 
1715 

36 
Brouwer Fixed Point Theorem 
1910 

37 
The Solution of a Cubic 
1500 

38 
Arithmetic Mean/Geometric Mean (Proof by Backward Induction) (Polya Proof) 
? ? 

39 
Solutions to Pell’s Equation 
1759 

40 
Minkowski’s Fundamental Theorem 
1896 

41 
Puiseux’s Theorem 
Victor Puiseux (based on a discovery of Isaac Newton of 1671) 
1850 
42 
Sum of the Reciprocals of the Triangular Numbers 
1672 

43 
The Isoperimetric Theorem 
1838 

44 
The Binomial Theorem 
1665 

45 
The Partition Theorem 
1740 

46 
The Solution of the General Quartic Equation 
1545 

47 
The Central Limit Theorem 
? 
? 
48 
Dirichlet’s Theorem 
1837 

49 
The CayleyHamilton Thoerem 
1858 

50 
The Number of Platonic Solids 
400 B.C. 

51 
Wilson’s Theorem 
1773 

52 
The Number of Subsets of a Set 
? 
? 
53 
Pi is Trancendental 
1882 

54 
Konigsberg Bridges Problem 
1736 

55 
Product of Segments of Chords 
300 B.C. 

56 
The HermiteLindemann Transcendence Theorem 
1882 

57 
Heron’s Formula 
75 

58 
Formula for the Number of Combinations 
? 
? 
59 
The Laws of Large Numbers 
<many> 
<many> 
60 
Bezout’s Theorem 
? 

61 
Theorem of Ceva 
1678 

62 
Fair Games Theorem 
? 
? 
63 
Cantor’s Theorem 
1891 

64 
L’Hopital’s Rule 
1696? 

65 
Isosceles Triangle Theorem 
300 B.C. 

66 
Sum of a Geometric Series 
260 B.C.? 

67 
e is Transcendental 
1873 

68 
Sum of an arithmetic series 
Babylonians 
1700 B.C. 
69 
Greatest Common Divisor Algorithm 
300 B.C. 

70 
The Perfect Number Theorem 
300 B.C. 

71 
Order of a Subgroup 
1802 

72 
Sylow’s Theorem 
1870 

73 
Ascending or Descending Sequences 
Paul Erdos and G. Szekeres 
1935 
74 
The Principle of Mathematical Induction 
1321 

75 
The Mean Value Theorem 
1823 

76 
Fourier Series 
1811 

77 
Sum of kth powers 
1713 

78 
The CauchySchwarz Inequality 
1814? 

79 
The Intermediate Value Theorem 
1821 

80 
The Fundamental Theorem of Arithmetic 
300 B.C. 

81 
Divergence of the Prime Reciprocal Series 
1734? 

82 
Dissection of Cubes (J.E. Littlewood’s ‘elegant’ proof) 
R.L. Brooks 
1940 
83 
The Friendship Theorem 
Paul Erdos, Alfred Renyi, Vera Sos 
1966 
84 
Morley’s Theorem 
1899 

85 
Divisibility by 3 Rule 
? 
? 
86 
Lebesgue Measure and Integration 
1902 

87 
Desargues’s Theorem 
1650 

88 
Derangements Formula 
? 
? 
89 
The Factor and Remainder Theorems 
? 
? 
90 
Stirling’s Formula 
1730 

91 
The Triangle Inequality 
? 
? 
92 
George Pick 
1899 

93 
The Birthday Problem 
? 
? 
94 
The Law of Cosines 
1579 

95 
Ptolemy’s Theorem 
120? 

96 
Principle of Inclusion/Exclusion 
? 
? 
97 
Cramer’s Rule 
1750 

98 
Bertrand’s Postulate 
1860? 

99 
Buffon Needle Problem 
1733 

100 
Descartes Rule of Signs 
1637 