The VRAM doesn't affect the resolution. The relation is more of resolution affecting VRAM usage. Sames for goes for textures. AA will also chew up VRAM (along with GPU processing power).
Not all VRAM is equal. Some lower end cards use DDR3 on the card, while the current cutting edge use GDDR5.
The memory bandwidth can also affect performance (more of how much VRAM the card can effectively use). You will find values such as 128, 256, 368.