Einstein is well known for his work in physics, but what about math? Did he brought new ideas to mathematics, or was he influenced by mathematical ideas? Before answering these questions, lets look on a short summary of Einstein achievements - from Wikipedia:
Albert Einstein was a German-born theoretical physicist. He is best known for his theory of relativity and specifically mass–energy equivalence, E = mc^2. Einstein received the 1921 Nobel Prize in Physics "for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect."
Einstein's many contributions to physics include his special theory of relativity, which reconciled mechanics with electromagnetism, and his general theory of relativity, which extended the principle of relativity to non-uniform motion, creating a new theory of gravitation. His other contributions include relativistic cosmology, capillary action, critical opalescence, classical problems of statistical mechanics and their application to quantum theory, an explanation of the Brownian movement of molecules, atomic transition probabilities, the quantum theory of a monatomic gas, thermal properties of light with low radiation density (which laid the foundation for the photon theory), a theory of radiation including stimulated emission, the conception of a unified field theory, and the geometrization of physics.
As you can see he contributed to a lot of different topics in physics. Math is mentioned nowhere in this short summary and for a good reason - he was after all a physicist. However to do anything in any of the fields mentioned above, a very good understanding of math is required.
In my opinion, Einstein relationship with math can be divided into two periods. The first period was in childhood when he first heard about mathematics:
Einstein was introduced to math when he was ten years old. In 1889, family friend Max Talmud, introduced him to mathematics, and philosophy texts, including Kant's Critique of Pure Reason and Euclid's Elements (Einstein called it the "holy little geometry book"). From Euclid, Einstein began to understand deductive reasoning, and by the age of twelve, he had learned Euclidean geometry. Soon thereafter he began to investigate calculus.
In 1895 Einstein tried to apply to Swiss Federal Institute of Technology in Zürich, Switzerland. He was required to pass an entrance exam which he failed - but he got exceptional marks in mathematics and physics. In 1896 he finally enrolled in the mathematics program at ETH. He graduated in 1900 from there with a degree in physics as a teacher of mathematics and physics.
His graduation marks the beginning of a second period - now he was no longer studying math. He was now looking on it as a tool to solve problems that he encountered. However math refused to be only a tool for him:
By mid 1901 he had a temporary job as a teacher, teaching mathematics at the Technical High School in Winterthur, and after this in a private school in Schaffhausen. These positions were all temporal - he had a desire to get to an university. It took a lot of time and effort but in 1912, after publishing a lot of groundbreaking articles (in physics), Einstein returned to Switzerland to accept a professorship at the ETH. There he met mathematician Marcel Grossmann who introduced him to Riemannian geometry, and at the recommendation of Italian mathematician Tullio Levi-Civita, Einstein began exploring the usefulness of general covariance (essentially the use of tensors) for his gravitational theory. Ultimately, the ideas and approaches of Riemann geometry provided him what he needed to develop his general relativity theory - the distortion of space-time under the influence of gravity. While it is usually thought that physics "push" mathematics in this particular moment it was mathematics that contributed to advance of physics.
The above is only a short summary - it is possible to say more, but the answers to the questions I started with are already obvious: While there are no evidence to suggest that Einstein had any direct influence on the development of math, he and his ideas were greatly effected by it. On the other hand, as it is always with physics, his theories put forward new problems that required development of new mathematical methods to be solved.
Quotes about math by Einstein:
"God does not care about our mathematical difficulties. He integrates empirically."
"Do not worry about your difficulties in Mathematics. I can assure you mine are still greater."
"As far as the laws of mathematics refer to reality, they are not certain, as far as they are certain, they do not refer to reality."
"Yes, we have to divide up our time like that, between our politics and our equations. But to me our equations are far more important, for politics are only a matter of present concern. A mathematical equation stands forever."
"Pure mathematics is, in its way, the poetry of logical ideas."
"How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?"
"One reason why mathematics enjoys special esteem, above all other sciences, is that its laws are absolutely certain and indisputable, while those of other sciences are to some extent debatable and in constant danger of being overthrown by newly discovered facts."
