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Having a higher resolution means the graphics card needs more memory to hold the color data for each pixel. How much is required depends on the color depth of your screen. If it's 24-bit, that's 3 bytes per pixel. So for 1600x1200 @ 24 bits/pixel that's 5MB or so of video memory (3 * 1600 * 1200 = 5760000). That used to matter many years ago but nowadays it is more a function of your monitor, as you say. For 2D, video cards just receive a bitmap organized by the CPU and forward them to the monitor (monitor doesn't store any data, just rewrites the pixels n times every second). For 3D it's more complex, and GPU speed/GPU memory bus rate matter a lot more for those sorts of things, because the GPU actually does some work.
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