It all depends on what you want to do.
If you're not a gamer or 3d artist, then you don't have to think twice about it. System ram for sure, it's general purpose, and benefits almost any application. It allows you to do more heavy multitasking, etc,...
If you plan on playing games, or work in anything else 3d (3dsmax/maya/blender/etc) the story changes a bit. The amount of video ram will be a key factor on how much (3d) stuff you can show on the screen at the same time. (Amount of vertices, texture memory, etc,...) this is important for games, but also for anything else 3d. (Say you are doing detailed architectural design or something in a 3d application, you will start feeling it when your card is showing more geometry on the screen than it can handle) - Unfortunately, it gets more complicated in this arena. You also want to look at the gpu and memory clock rates on your video card. - If you plan to get a video card you should be looking at gpu and memory clock rate, and amount of memory.
Almost any modern video card, integrated or not, will be able to handle the native resolutions for most monitors today, so if you're not doing anything 3d i wouldn't even worry about it.