How does a graphics card work?

Complicated question, especially at the moment. You need to make a distinction between a video card/graphics card and GPU (graphics processing unit). If I assume you mean a graphics a card like AMD or Radeon or Nvidia GPUs then..

"It is a specialized electronic circuit designed to rapidly manipulate and alter memory to accelerate the creation of images in a frame buffer intended for output to a display. GPUs are used in embedded systems, mobile phones, personal computers, workstations, and game consoles. Modern GPUs are very efficient at manipulating computer graphics and image processing, and their highly parallel structure makes them more effective than general-purpose CPUs for algorithms where processing of large blocks of data is done in parallel. In a personal computer, a GPU can be present on a video card, or it can be on the motherboard or—in certain CPUs—on the CPU die."

Of course there are many different types, the main being dedicated, Integrated, hybrid, stream processing and general purpose GPUs (GPGPU) and External GPU (eGPU).

If you want a specific answer, please ask a specific question and I'll be happy to answer :)