Physicist: There are a lot of ways.
A fractal is just a structure that stays interesting no matter how far you zoom in. More than that, generally you can’t tell how far you’ve zoomed in (it looks the same at many different size scales). Making up new fractals is surprisingly easy.
The most famous one (that you’ll find on posters in dorm rooms) is the “Mandelbrot set”, followed by the “Sierpinski Triangle”. I haven’t taken a survey, but these seem to be the most popular.
It’s pretty straightforward how the Sierpinski triangle is made: make a triangle, put three triangles in its corners, put three triangles in those triangle’s corners, etc. This sort of idea: take a shape and then populate it with smaller versions of itself, is arguably the most common way to generate fractals.
The Mandelbrot set on the other hand involves complex numbers and computing power that wasn’t available until the ’80s.
To determine if a point is in the Mandelbrot set, start with the recursion:
z n+1 = z n 2 + c
z 0 =0
This means “square what you’ve got, add c, then take the result, square it, add c, then take the result, …”
For different values of c the string of numbers you get out does different things. For example;
for c = 1, you get: 0, 1, 2, 5, 26, 677, … (that’s 0, 1=02+1, 2=12+1, 5=22+1, …)
for c = -0.5, you get: 0, -0.5, -0.25, -0.4375, -0.30859375, -0.404769897, …
The Mandelbrot set is defined as the set of values of c that lead to strings of numbers that stay bounded. So, c=-0.5 is in the set because the string of numbers it makes stays in more or less the same place (it stays between -0.5 and 0 forever). But c=1 is not in the set because its string of numbers blows up.
It’s a little more complicated because you actually consider complex numbers (which is why the picture you get is in a plane).
Finally, the set itself isn’t terribly interesting, but its boundary is. The boundary between what’s in the set and what’s not (what generates a string of numbers that behaves, as opposed to a string that blows up) is infinitely squiggly.
Almost every time you see the Mandelbrot set, colors are included that indicate how fast the strings generated by numbers not in the set blow up. Also, without colors it’s boring.
Another classic is the Dragon Curve:
The Dragon Curve is what you get what you fold a piece of paper in the same direction over and over (forever), and then unfold it at 90° angles.
Long story short: there are many ways to create fractal patterns, but it’s not always easy to guess what technique will lead to a fractal until you try it.