Fractal geometry is the study of any shape with some degree of self-similarity, like broccoli, or mountains, or the Sierpinski Triangle. Self-similarity is the property that a small part of the whole looks like the whole (only smaller of course)! This is true of broccoli, for example. Examining a head of broccoli, you may notice that if you tear off a piece, it looks like a miniature head of broccoli! And if you tear off a piece of this piece it looks like a tinier head of broccoli! Ditto for the mountain. Observing a mountain from far away, you see peaks and valleys. Come a little closer and you will see peaks and valleys on the peaks and valleys! And on these smaller peaks and valleys, you may find yet smaller peaks and valleys. In other words, a small portion of the whole has the same general shape as the whole!
With physical objects like broccoli and mountains, after a while of looking at smaller and smaller parts of the whole, the small parts DO NOT look like the whole. But "true" fractals continue to look like the whole no matter how small the part you observe is. For example, draw a triangle. Then connect the midpoints of the three sides, forming a small triangle inside the original triangle. Notice that there are now actually 4 smaller triangles! For each of the three triangles outside the "middle" triangle, connect the midpoints of the three sides, forming another (still smaller) triangle inside surrounded by three other small triangles. Then connect the midpoints of the three sides of each of these surrounding triangles, and so forth. The Sierpinski Triangle is what you get if you continue this process FOREVER. Make your own fractals!
Check out the self-similarity (repetition of pattern at different scales) in these fractals created by Newton's Method. Click on the image for further details.