For basic, entry-level 1080p gaming, the minimum amount of  VRAM (video RAM) that any graphics card should have is 4 GB. You'll notice, though, that all of our recommended cards have 8 GB or more, and that's because today's big-budget 3D games have graphics that can eat up more than 6 GB.
Once you're gaming at 1440p or higher, and using high graphics settings, 6GB just won't cut the mustard and even 8 GB might be a bit limiting in some games. This is why the latest mid-range graphics cards, between $350 and $600 in price, sport 12 or 16 GB of VRAM.
The latter is more than enough for almost any game. You'll only need more than this in a small number of games and even then, only under very specific circumstances, such as 4K with maximum settings and frame generation enabled.
If you're using your GPU to do content creation or AI workloads, though, the more VRAM it has, the better the graphics card will be able to handle those tasks.