Chronology

Russell and Whitehead publish the first volume of Principia Mathematica. They attempt to put the whole of mathematics on a logical foundation. They were able to provide detailed derivations of many major theorems in set theory, finite and transfinite arithmetic, and elementary measure theory. The third and final volume will appear three years later, while a fourth volume on geometry was planned but never completed.
Steinitz gives the first abstract definition of a field in Algebraische Theorie der Körper.

Sergi Bernstein introduces the "Bernstein polynomials" in giving a constructive proof of Weierstrass's theorem of 1885.

Hardy receives a letter from Ramanujan. He brings Ramanujan to Cambridge and they go on to write five remarkable number theory papers together.
Weyl publishes Die Idee der Riemannschen Flache which brings together analysis, geometry and topology.

Hausdorff publishes Grundzüge der Mengenlehre in which he creates a theory of topological and metric spaces.
Bieberbach introduces the "Bieberbach polynomials" which approximate a function that conformally maps a given simply-connected domain onto a disc.
Harald Bohr and Edmund Landau prove their theorem on the distribution of zeros of the zeta function.

Einstein submits a paper giving a definitive version of the general theory of relativity. (See this History Topic.)

Bieberbach formulates the Bieberbach Conjecture.
Macaulay publishes The algebraic theory of modular systems which studies ideals in polynomial rings. It contains many ideas which today occur in the theory of "Grobner bases".
Sierpinski gives the first example of an absolutely normal number, that is a number whose digits occur with equal frequency in whichever base it is written.

Russell publishes Introduction to Mathematical Philosophy which had been largely written while he was in prison for anti-war activities.
Hausdorff introduces the notion of "Hausdorff dimension", which is a real number lying between the topological dimension of an object and 3. It is used to study objects such as Koch's curve.