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Fractal
Informally speaking fractal is a shape that's geometrically "infinitely complex" while being described in an extremely simple way, e.g. with a very simple formula. Shapes found in the nature, such as trees, mountains or clouds, are often fractals. Fractals show self-similarity, i.e. when "zooming" into an ideal fractal we keep seeing it is composed, down to an infinitely small scale, of shapes that are similar to the shape of the whole fractal; e.g. the branches of a tree look like smaller versions of the whole tree etc.
Fractals are the beauty of mathematics, they can impress even complete non-mathematicians and so are probably good as a motivational example in math education.
Fractal is formed by iteratively or recursively (repeatedly) applying its defining rule -- once we repeat the rule infinitely many times, we've got a perfect fractal. In the real world, of course, both in nature and in computing, the rule is just repeat many times as we can't repeat literally infinitely. The following is an example of how iteration of a rule creates a simple tree fractal; the rule being: from each branch grow two smaller branches.
V V V V
\ / \ / V \ / \ / V
| | _| | | |_ >_| | | |_<
'-.| |.-' '-.| |.-' '-.| |.-'
\ / \ / \ / \ /
\ / \ / \ / \ /
| | | |
| | | |
| | | |
iteration 0 iteration 1 iteration 2 iteration 3
Mathematically fractal is a shape whose Hausdorff dimension (the "scaling factor of the shape's mass") is non-integer. For example the Sierpinski triangle can normally be seen as a 1D or 2D shape, but its Hausdorff dimension is approx. 1.585 as if we scale it down twice, it decreases its "weight" three times (it becomes one of the three parts it is composed of); Hausdorff dimension is then calculated as log(3)/log(2) ~= 1.585.
L-systems are one possible way of creating fractals. They describe rules in form of a formal grammar which is used to generate a string of symbols that are subsequently interpreted as drawing commands (e.g. with turtle graphics) that render the fractal. The above shown tree can be described by an L-system. Among similar famous fractals are the Koch snowflake and Sierpinski Triangle.
/\
/\/\
/\ /\
/\/\/\/\
/\ /\
/\/\ /\/\
/\ /\ /\ /\
/\/\/\/\/\/\/\/\
Sierpinski Triangle
Fractals don't have to be deterministic, sometimes there can be randomness in the rules which will make the shape be not perfectly self-similar (e.g. in the above shown tree fractal we might modify the rule to from each branch grow 2 or 3 new branches).
Another way of describing fractals is by iterative mathematical formulas that work with points in space. One of the most famous fractals formed this way is the Mandelbrot set. It is the set of complex numbers c such that the series z_next = (z_previous)^2 + c, z0 = 0 does not diverge to infinity. Mendelbrot set can nicely be rendered by assigning each iteration's result a different color; this produces a nice colorful fractal. Julia sets are very similar and there is infinitely many of them (each Julia set is formed like the Mandelbrot set but c is fixed for the specific set and z0 is the tested point in the complex plain).
Fractals can of course also exist in 3 and more dimensions so we can have also have animated 3D fractals etc.
Fractals In Tech
Computers are good for exploring and rendering fractals as they can repeat given rule millions of times in a very short time. Programming fractals is quite easy thanks to their simple rules, yet this can highly impress noobs.
3D fractals can be rendered with ray marching and so called distance estimation. This works similarly to classic ray tracing but the rays are traced iteratively: we step along the ray and at each step use an estimate of the current point to the surface of the fractal; once we are "close enough" (below some specified threshold), we declare a hit and proceed as in normal ray tracing (we can render shadows, apply materials etc.). The distance estimate is done by some clever math.
Mandelbulber is a free, advanced software for exploring and rendering 3D fractals using the mentioned method.
Marble Racer is a FOSS game in which the player races a glass ball through levels that are animated 3D fractals. It also uses the distance estimation method implemented as a GPU shader and runs in real-time.
Fractals are also immensely useful in procedural generation, they can help generate complex art much faster than human artists, and such art can only take a very small amount of storage.
There also exist such things as fractal antennas and fractal transistors.