Short answers, pulled from the story.
Space is a three-dimensional continuum containing positions and directions, and it is one of the few fundamental quantities in physics. It cannot be defined through other quantities because nothing more fundamental is known, though it can be explored through measurement and experiment.
Newton used a spinning bucket of water to argue that space exists independently of matter. Water hung from a rope develops a concave surface as it spins, and the curve remains even after the bucket stops, which Newton said proves the effect comes from motion relative to space itself rather than relative to the bucket.
Isaac Newton viewed space as absolute, existing permanently and independently of any matter within it. Gottfried Leibniz held that space is only the collection of spatial relations between objects, describing it as that which results from places taken together.
Non-Euclidean geometry describes space as curved rather than flat by dropping Euclid's parallel postulate. Around 1830 the Hungarian Janos Bolyai and the Russian Nikolai Ivanovich Lobachevsky separately published hyperbolic geometry, and in the 1850s Bernhard Riemann developed elliptical geometry.
In 1905 Albert Einstein published special relativity, combining space and time into spacetime. His general theory of relativity then described gravity as a curving of spacetime near significant masses, replacing the idea of gravity as a force acting within space.
LIGO scientists reported the first direct observation of gravitational waves on the 14th of September 2015. Indirect evidence had earlier been found in the motions of the Hulse-Taylor binary system.
Space appears to have been created in the Big Bang, 13.8 billion years ago, and has been expanding ever since. Its overall shape is not known, but it is expanding very rapidly due to cosmic inflation.