What is the algorithm employed by video cards when one talks about for example
I thought the algorithm was about looking at neighboring pixels for correction. Yet, I hear that the implementation is just upsampling followed by downsampling.
There are various algorithms to do anti-aliasing. I think that the one that you mean is called super-sampling. The idea is that the scene is rendered in x8 resolution of what you eventually see. Then it is low-pass filtered, to eliminate aliasing of higher frequencies in the next step, which is down-sampling x8.
This is described in detail here.