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Euclid's parallel postulate is the assumption that distinguishes Euclidean geometry from other geometries.
It states that if two lines are cut by a transversal, and the same-side interior angles add up to less than 180 degrees, then the two lines intersect on the same side of the transversal.
It turns out that many statements are actually equivalent to Euclid's fifth postulate. This means that in other systems of geometry, these statements are no longer true. In addition, had we started with one of these statements as Euclid's fifth postulate instead, we could (in principle) prove Euclid's parallel postulate and call that a theorem instead. Here are some (of course not all) of those statements:
(Playfair's Axiom) Given a line and a point not on the line, there exists exactly one line through the point that is parallel to the given line.
(Pythagorean Theorem) In a right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse.
There exist two triangles that are similar but not congruent.
The sum of the interior angles of a triangle is 180 degrees.
Every triangle has a circumcircle.
(Wallis Axiom) There is no upper limit to the area of a triangle.
If three angles of a quadrilateral are all right angles, then the fourth angle is also a right angle.
These equivalent statements were a lot of trouble for mathematicians because some of the statements, such as Playfair's Axiom, seemed self-evident. Many mathematicians were unhappy with Euclid's fifth postulate because it was deemed much less obvious than his other postulates, so they set out in an effort to prove it from the other four. Many false proofs were found in which the author had subconsciously made an assumption that seemed self-evident but was, in fact, equivalent to the parallel postulate that they were trying to prove. Only after a long time was it finally proven that Euclid's fifth postulate cannot be proven from the other four.