Euclid's postulates
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Around 300 BCE, Euclid wrote the Elements, the most influential textbook in history, by building all of geometry from a short list of starting assumptions. His five postulates are the rules of construction you are allowed to take for granted — the moves a straightedge and compass can make — from which every theorem must then be proved.
In plain modern wording the five are: (1) you can draw a straight segment from any point to any other point; (2) you can extend any straight segment as far as you like in a straight line; (3) you can draw a circle with any centre and any radius; (4) all right angles are equal to one another; and (5) the parallel postulate — if a line crossing two others makes the two interior angles on one side add to less than two right angles, those two lines, extended, meet on that side. Euclid also lists 'common notions', more general truths like 'things equal to the same thing are equal to each other'.
The first four postulates are short, intuitive, and were never seriously doubted. The fifth is wordier and felt less obvious, and for two thousand years mathematicians tried to derive it from the other four — and failed. We now know why: the parallel postulate is independent of the rest, so it can be assumed true (Euclidean geometry) or replaced (hyperbolic and elliptic geometries), each giving a consistent geometry. The fifth postulate was never wrong; it simply was not forced, and recognising that opened the door to non-Euclidean geometry.
Postulate 3 lets you set a compass to any radius and swing a full circle; postulate 1 lets you join two marked points with a segment. Together they justify the very first construction in the Elements: building an equilateral triangle on a given segment by drawing two circles and connecting their crossing point to the endpoints.
The postulates are the legal moves; every theorem is a proof using only those moves.
The parallel postulate was never disproved — it is independent of the others, so consistent geometries exist both with it and without it. Non-Euclidean geometry adds options; it does not refute Euclid.