Technically, a “fractal” is any entity with a fractional dimension. One way to measure this by how its area* multiplies when you scale it up or down. A line that’s twice as long has 2x the area. A square twice as wide has 4x the area. A cube has 8x. This implies the formula scaleFactor = 2^dimension, or dimension = log-base-2(scaleFactor). The Serpinski Triangle is a fractal that contains 3 copies of itself, each at half scale; so if you scale one to be twice as wide, it’s equivalent to multiplying the area by 3. From our formula earlier, this means its dimension is log-base-2 of 3, or about 1.585-- somewhere between 1 and 2 dimensional!

Note that the Serpinski Triangle is made of copies of itself-- this makes it a “self-similar” fractal, which ironically makes it easier to work with. This is what people generally think of when they say “fractal”, and has essentially become the common usage of the term. But note that technically, not all self-similar shapes are fractal (a square can be made of 4 scaled squares), and interestingly, not all fractal shapes are self-similar! Measuring their dimension can be harder, but in your example of eg. the British coastline, notice how the scale at which you measure things changes the length of the coastline. Do you measure each cove? Each tiny protrusion of rock? Each individual grain of sand as the water of the ocean wraps around it? You can compare your answers at different scales and (somehow) use that to calculate a fractional dimension, since they’ll scale differently than a flat surface coastline would.

* there’s a general name for length/area/volume/etc. which I should be using but I forgot what it is

Edit: Almost forgot to answer your second question; they’re an approximation. Computers simulate fractals similarly to how they compute irrational numbers like pi, where they only calculate up to a certain decimal point. For rendering a self-similar fractal, this means they render a certain number of smaller copies, where anything beyond the smallest copy is simply assumed to be in or out of the fractal by default.