Short answers, pulled from the story.
Euclidean geometry is attributed to Euclid, an ancient Greek mathematician. He described the system in his textbook Elements, which organized earlier geometric knowledge into the first complete axiomatic system.
Elements contains 13 books. Books I-IV and VI cover plane geometry; Books V and VII-X address number theory using geometric methods; Books XI-XIII treat solid geometry, including the Platonic solids.
The parallel postulate states that if a line crosses two other lines and creates interior angles on one side summing to less than two right angles, those two lines will eventually meet on that side. Euclid reserved its use until his 29th proposition, suggesting he considered it qualitatively different from the other four postulates.
Albert Einstein's theory of general relativity shows that gravity curves space, so a triangle formed by three rays of light does not have interior angles summing to 180 degrees. Euclidean geometry is a good approximation only over short distances and in weak gravitational fields.
Pierre Wantzel published the proof in 1837. Compass-and-straightedge constructions involve equations whose order is an integral power of two, which rules out the trisection problem as well as doubling the cube.
Einstein predicted that gravity would bend rays of light, meaning space does not follow Euclidean rules. Observations during a solar eclipse in 1919 confirmed the slight bending of starlight by the Sun, and corrections derived from non-Euclidean geometry are now part of the software running the GPS system.