Question: Isn't it just a pixil anyway?

All right, I need a favour. I've been running through this pixels and vectors thing in my head and I just can't manage to wrap my brain around a vector. Can anyone give me layman's terms of what this is, because I can understand how a computer can see and grid and go "Square A11 = Bright Yellow" but I don't quite get the idea of a block of colour or shape in a spatial context. Does it have co-ordinates? And if we view it on a screen isn't it a pixel image anyway? So if there's no way to encounter it in its pure form then can it really be said to be an alternative to a pixel-based image? In short: Help!