A shader is code the graphics card runs massively in parallel: once per vertex to position geometry, once per pixel to color it. That parallelism is why a laptop can render millions of pixels of animated imagery at 60 frames per second.
For visual artists, shaders are the direct line to the machine. A fragment shader is a formula for what is here, given position and time, and entire scenes (clouds, fractals, oceans) fit in a page of code. Communities like Shadertoy made the form a genre.
Nearly everything in a modern visualizer is shaders end to end. Photism’s scenes and its lens effects are shader programs steered by the audio analysis.