The Parallel Postulate: Given a line, and a point not on that line, there is one and only line that 1) passes through that point and 2) is parallel to the other line.
This is a foundation of Euclidean geometry, Euclid's fifth postulate (Actually I find it's a version of Playfair's Axiom, which is equivalent). It is, however, not necessarily so; coherent geometrical systems have been constructed under alternate assumptions, e.g.:
Hyperbolic geometry, or Lobachevskian geometry (or even Lobachevsky-Bolyai-Gauss geometry), says that there may be many lines that pass through our point and do not intersect our line.
Elliptic geometry, or Riemannian geometry, says that there are no such lines.
An example of the differences these systems may make in the "real world": Euclidean triangles have 180 degrees, Riemannian triangles have more than 180 degrees, and Lobachevskian triangles have less than 180 degrees, the difference depending on the size of the triangle.