They are all around us in the form of nature. Good examples of fractals in nature are the shape of a mountain range, the winding of a coastline, cloud formations, even the formations of a flickering fire. It comes from the Latin fractus . The verb frangere means to break, to create irregular fragments. This area of Math falls into the field called Chaos or Dynamical Systems.
Fractals are simple, nonlinear systems with complex dynamical properties.
That is, the shape repeats over and over to what looks like infinity. The other is that on the boundary the process is very chaotic. It contains chaotic sequences of points that never approach any kind of regularity. Both fractals and chaos are branches of math that are currently getting a lot of attention for they are both modern discoveries. They are an area of geometry that parts from the standard Euclidian straight-line geometry.