VRAM (pronounced "vee ram") stands for video random access memory, or video RAM. This is a special type of memory used on some video adapters to speed up the display of images on the screen. VRAM costs more than regular RAM (DRAM, dynamic RAM), but it does make the screen snappier.
Whichever type of RAM your video adapter uses, the amount of memory determines how many colors you can see on the screen: the more video RAM you have, the more colors your monitor can display and the higher the resolution will be.
Analog monitors like the VGA can display an infinite number of colors, they say, because they can vary the intensity of each color gun over an infinite range. It's the adapter circuits that determine how many colors are displayed. But anyway, that's not the main point of VRAM.