Hyperbolic geometry is a non-Euclidean geometry that explores spaces where the parallel postulate of Euclidean geometry does not hold. In this type of geometry, through a point not on a given line, there are infinitely many lines that do not intersect the given line. This leads to unique properties, such as the sum of angles in a triangle being less than 180 degrees. One common model used to visualize hyperbolic geometry is the Poincaré disk model, where the entire hyperbolic plane is represented within a circle. Another model is the hyperboloid model, which uses a three-dimensional surface to illustrate hyperbolic space. These models help in understanding the complex relationships and structures within hyperbolic geometry.