Non-Euclidean geometry is a type of geometry. Non-Euclidean geometry only uses some of the "postulates" (assumptions) that Euclidean geometry is based on. In normal geometry, parallel lines can never meet. In non-Euclidean geometry they can meet, either once (elliptic geometry), or infinity many (hyperbolic geometry) times.
An example of Non-Euclidian geometry can be seen by drawing lines on a ball or other round object, straight lines that are parallel at the equator can meet at the poles
It is called "Non-Euclidean" because it is different from Euclidean geometry, which was discovered by an Ancient Greek man named Euclid. The different names for non-Euclidean geometries come from thinking of "straight" lines as curved lines, either curved inwards like an ellipse, or outwards like a hyperbola.