There are various definitions of what a fractal is. One of the easiest is that a fractal is usually self-similar. That means that it repeats itself. For an example, look at the following fractal.
This is a Van Koch fractal. It is based on a very simple shape.
To create the fractal, the flat lines are replaced by the entire shape itself.
This process is repeated again and again to create an infinitely complicated fractal. Still, every part of the fractal contains the original shape. We say that the fractal is self-similar. Most fractals in Ultra Fractal are calculated differently, but the principle of self-similarity still applies.
This is also the reason that it is so popular to zoom into fractals: there are always more details to be discovered, no matter how far you zoom.
Next: Julia sets
What are fractals?