In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate that defines Euclidean geometry isn't true. On a hyperbolic plane, lines that started out parallel will become further and further apart.
Replacing this rule means that hyperbolic geometry acts differently from ordinary flat plane geometry. For example, triangles will have angles that add up to less than 180 degrees, meaning that they are too pointy and will look like the sides are sinking into the middle.
Many real objects look like hyperbolic planes. For example, some types of coral and lettuce are shaped like pieces of hyperbolic planes. Other people say that it's easier to draw a map of the Internet when your map isn't flat, because there are a lot of computers around the edges but very few in the center. Some physicists even think our universe is a little bit hyperbolic.