Non Euclidean geometry is geometry that refutes Euclid's fifth postulate: Given a point p and a line l, there is exactly one line through p that is parralel to l.
Other than this difference, non-Euclidean geometry holds to the rest of Euclid's postulates:
Now, there are two ways that you can refute this.
One way is to say: Given a line l and point p, there are no lines parralel to l through p. This leads to Spherical/Elliptical Geometry.
The other is to say: Given a line l and a point p, there are infinite lines through p parralel to l. This leads to Hyperbolic Geometry