It depends what you're coding in and what you're making. If, for example, you're making a 2D game in DBP and making a border that goes around the outside of the screen, then it's all about resizing a high res graphics.
You'd want to make one version of the graphic in the highest resolution you'll support (say 1600x1200), and then if the game is running in a lower resolution, when it loads up you'll need to scale the graphic down.
If you want to support wide and 4:3 screen, you'd need to make separate graphics for the sides and top/bottom of the border. The sides could just be resized as mentioned above, but the top/bottom graphics would need to be made in such away that they could be pasted to the screen completely in widescreen, but with some of the graphic running off the screen in 4:3 resolution, but such that it still looks right. Or you could stretch/squash the bottom graphic.
Supporting multiple resolutions, hardware speed etc. is definitely one of the more annoying tasks we face as game developers. I'm doing a lot of Android dev now, and have to support 4 different pixel densities, about 10 different resolutions, and 3 different aspect ratios. It's a pain in the arse, but it's something you need to consider up front before you make any graphics, and design them around the concept of resizing.
Good luck!
