What is fwidth and how does it work? The OpenGL documentation states that fwidth returns the sum of the absolute value of derivatives in x and y.

What does this mean in less mathematical terms, and is there a way to visualize it?

Based on my understanding of the function, fwidth(p) has access to the value of p in neighboring pixels. How does this work on the GPU without drastically impacting performance, and does it work reliably and uniformly across all pixels?
Pixel screen-space derivatives do drastically impact performance, but they impact performance whether you use them or not, so from a certain point of view they're free!

Every GPU in recent history packs a quad of four pixels together and puts them in the same warp/wavefront, which essentially means that they're running right next to each other on the GPU, so accessing values from them is very cheap. Because warps/wavefronts are run in lockstep, the other pixels will also be at exactly the same place in the shader as you are, so the value of p for those pixels will just be sitting in a register waiting for you. These other three pixels will always be executed, even if their results will be thrown away. So a triangle that covers a single pixel will always shade four pixels and throw away the results of three of them, just so that these derivative features work!

This is considered an acceptable cost (for current hardware) because it isn't just functions like fwidth that use these derivatives: every single texture sample does as well, in order to pick what mipmap of your texture to read from. Consider: if you are very close to a surface, the UV coordinate you are using to sample the texture will have a very small derivative in screen space, meaning you need to use a larger mipmap, and if you are farther the UV coordinate will have a larger derivative in screen space, meaning you need to use a smaller mipmap.

As far as what it means in less mathematical terms: fwidth is equivalent to abs(dFdx(p)) + abs(dFdy(p)). dFdx(p) is simply the difference between the value of p at pixel x+1 and the value of p at pixel x, and similarly for dFdy(p).