I made a couple of vector shapes in AI and I get this strange thin border appearing around them. I can't get rid of it. Any suggestions?
Adobe Illustrator constructs objects with vector mathematics. However, it has to interpret that vector data into pixels in order to display it on the (pixel-based) monitor.
To create smooth lines on screen it anti-aliases pixels when two colors lie next to each other. It essentially "blends" the two colors together over a pixel or two in order to represent their position.
Sometimes this anti-aliasing isn't entirely smooth if your monitor is not capable of high quality rendering. Therefore you get a small, off-color, hairline, edge with some objects.
This hairline will generally disappear if you uncheck Anti-Alias Artwork in the preferences. It will also disappear if you print the artwork or export artwork using the "Art Optimized" anti-alias setting.
This small anti-aliasing pixel is generally nothing to be concerned about.