Video card and graphics card are terms used interchangeably.
Video memory is an array of physical chips on the graphics card, the graphics card has it's own graphics processing unit (GPU for short) which is a bit like a CPU except that it's specialised for doing graphics instructions. The GPU requires its own dedicated video memory to read/write from because we want as much performance as possible it needs its own dedicated memory.
The more memory a video card has the more art assets it can store and use at any one time and that helps developers make games look better.
Graphics cards can look vastly different from one another primarily due to the range of cooling they require, some are small with small passive coolers and others are huge (up to a foot long) and require massive heatsinks and multiple fans. Just google image search video cards to get an idea.