An hyperbolic surface is a saddle-shaped surface whose edges also curve upwards. If we took a piece of paper and gently folded it in half to create a very shallow taco shell and then gently folded that taco shell backwards (pull the open ends back towards each other) that paper would resemble an hyperbolic surface. When multiple hyperbolic surfaces are seen together, the cluster closely resembles a coral reef.
Hyperbolic geometry occurs on these surfaces. Imagine you want to plot a line on a regular, Euclidean plane. That line would be straight, we could visually see that it was. If you plot that same line on an hyperbolic surface, it would appear curved because the surface on which it is plotted is curved.
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