When I was a teenager, I was given the book Men of Mathematics by E. T. Bell, and I rather enjoyed it. I know that this book has been criticized for various reasons and I might even agree with some of the criticism, but let's not digress onto that. E. T. Bell made a reasonable list of 34 of the greatest mathematicians from the ancient Greek period to the end of the 19th century. His list isn't perfect — maybe he should have included Klein or skipped Poncelet — but no such list can be perfect anyway. I think that his selection was good. He also made the careers of these mathematicians exciting and he addressed why mathematicians care about them today. It helped me learn what achievements and topics in mathematics are important.
(Just to head off discussion, the standard complaints include that the title is sexist and so is the book, that Bell was loose with biographical facts, and that he "chewed the scenery".)
After the period covered by Bell, there is a 50-70 year gap, followed by the the Fields Medals. The list of Fields Medalists has its own limitations, but it is another interesting, comparably long list of great mathematicians. Somewhat accidentally, this list also orients and motivates advanced mathematics students today.
But who are the great mathematicians in the gap itself, that is, those born between 1850 and 1900, or say between 1860 and 1910? (Or 1920 at the latest; that is what I had before.) There is a good expanded list of mathematicians with biographies at the St. Andrews site. However, it is too long to work as a sequel to Bell's book. (Not that I plan to write one; I'm just asking.) If you were to make a list of 20 to 50 great mathematicians in this period, how would you do it or who would they be? Presumably it would include Hilbert, but who else? (Poincaré was born in 1854 and is the latest born in Bell's list.)
For example, I know that there was the IBM poster "Men of Modern Mathematics" that also earned some criticism. But I don't remember who was listed, and my recollection is that the 1850-1900 period was somewhat cramped.
I decided based on the earliest responses to convert this list to community wiki. But I am not just asking people to throw out names one by one and then vote them up. If a good reference for this question already exists, or if there is some kind of science to make a list, then that would be ideal. If you would like to post a full list, great. If you would like to list one person who surely should be included and hasn't yet been mentioned, then that's also reasonable. It may be better to add years of birth and death in parentheses, for instance "Hilbert (1862-1943)".
Note: To make lists, you should either add two spaces to the end of each line, or " – " (space dash space) to the beginning of each line.
Best Answer
The St. Andrews site is an invaluable resource. From that list, I picked (usually) at most one great mathematician born in each year from 1860 to 1910:
$\textbf{EDIT: By popular demand, the list now extends from 1849 to 1920.}$
1849: Felix Klein, Ferdinand Georg Frobenius
1850: Sofia Vasilyevna Kovalevskaya
1851: honorable mention: Schottky
1852: William Burnside
1853: honorable mentions: Maschke, Ricci-Curbastro, Schoenflies
1854: Henri Poincare
1856: Emile Picard (honorable mention: Stieltjes)
1857: honorable mention: Bolza
1858: Giuseppe Peano (honorable mention: Goursat)
1859: Adolf Hurwitz (honorable mention: Holder)
1860: Vito Volterra
1861: honorable mention: Hensel
1862: David Hilbert
1864: Hermann Minkowski
1865: Jacques Hadamard (honorable mention: Castelnuovo)
1868: Felix Hausdorff
1869: Elie Cartan
1871: Emile Borel (honorable mentions: Enriques, Steinitz, Zermelo)
1873: honorable mentions: Caratheodory, Levi-Civita, Young
1874: Leonard Dickson
1875: Henri Lebesgue (honorable mentions: Schur, Takagi)
1877: Godfrey Harold Hardy
1878: Max Dehn
1879: honorable mentions: Hahn, Severi
1880: Frigyes Riesz
1881: Luitzen Egbertus Jan Brouwer
1882: Emmy Amalie Noether (honorable mentions: Sierpinski, Wedderburn)
1884: George Birkhoff, Solomon Lefschetz
1885: Hermann Weyl (honorable mention: Littlewood)
1887: Erich Hecke (honorable mentions: Polya, Ramanujan, Skolem)
1888: Louis Joel Mordell (honorable mention: Alexander)
1891: Ivan Matveevich Vinogradov
1892: Stefan Banach
1894: Norbert Wiener
1895: honorable mention: Bergman
1896: Carl Ludwig Siegel (honorable mention: Kuratowski)
1897: honorable mention: Jesse Douglas
1898: Emil Artin, Helmut Hasse (honorable mentions: Kneser, Urysohn)
1899: Oscar Zariski (honorable mentions: Bochner, Krull, Ore)
1900: Antoni Zygmund
1901: Richard Brauer
1902: Alfred Tarski (honorable mention: Hopf)
1903: John von Neumann (hm's: Hodge, Kolmogorov, de Rham, Segre, Stone, van der Waerden)
1904: Henri Cartan (honorable mentions: Hurewicz, Whitehead)
1905: Abraham Adrian Albert
1906: Kurt Godel, Andre Weil (honorable mentions: Dieudonne, Feller, Leray, Zorn)
1907: Lars Ahlfors, Hassler Whitney (honorable mentions: Coxeter, Deuring)
1908: Lev Pontrjagin
1909: Claude Chevalley, Saunders Mac Lane (honorable mentions: Stiefel, Ulam)
1910: Nathan Jacobson (honorable mention: Steenrod)
1911: Shiing-shen Chern (honorable mentions: Birkhoff, Chow, Kakutani, Witt)
1912: Alan Mathison Turing (honorable mentions: Eichler, Zassenhaus)
1913: Samuel Eilenberg, Paul Erdos, Israil Moiseevich Gelfand (dis/honorable mention: Teichmuller)
1914: honorable mentions: Dantzig, Dilworth, Kac
1915: Kunihiko Kodaira (honorable mentions: Hamming, Linnik, Tukey)
1916: Claude Elwood Shannon (honorable mention: Mackey)
1917: Atle Selberg (honorable mentions: Iwasawa, Kaplansky)
1918: Abraham Robinson
1919: honorable mention: Julia Robinson
1920: Alberto Calderon