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 infinitely many times (elliptic geometry), or never (hyperbolic geometry).
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 mathematician called Euclid. The different names for non-Euclidean geometries came from thinking of "straight" lines as curved lines, either curved inwards like an ellipse, or outwards like a hyperbola.
The first authors of non-Euclidean geometries were the Hungarian mathematician János Bolyai and the Russian mathematician Nikolai Ivanovich Lobachevsky, who separately published treatises on hyperbolic geometry around 1830. As well Eugenio Beltrami published book on non-Eucludean geometry in 1868. Others, such as Carl Friedrich Gauss, had earlier ideas, but did not publish their ideas at the time.