Filament has a natural stiffness, but when heated up to temperatures necessary for 3D printing, it gets significantly softer. ABS tends to be stiffer than PLA, so it is therefore better for printing overhangs. Still, don't be afraid to try extreme examples with both filaments. The Solidoodle is rated to 45 degrees of overhang, but skilled users have stretched a lot more from it.
There are some design choices you can make to help your overhangs print better, which are explained below. You can also try adding a fan to cool filament. The directions are on the fan installation page.
Design Without Overhangs
Sometimes you can avoid overhangs completely by making different design choices.
For example, you might encounter overhangs when you try to create an interior ceiling. You can then remove overhangs by designing a pyramidal or conical ceiling.
Change Your Model's Orientation
"When in doubt, turn it upside down."
Though this isn't strictly true, it is important to remember that your models can be printed in any direction. A model doesn't always have to print standing up like you find it real life. Something like a bookcase would have a lot of overhangs with each shelf, but printing it on its back eliminates all of those overhangs. If you don't have another flat plane to print on, you can still minimize the amount of support material would otherwise need, giving you a cleaner and easier print.
Span Distances with Bridges
Bridges are flat sections that span some open area. While bridges are still a type of overhang, they are a little easier for the printer to print. Because the printer can anchor the ends of a string of filament on either end of the bridge, the filament is less likely to droop (though it's still possible!) than if you had a section extending off into nowhere like a diving board. It is important to remember that these bridges need to be completely level with the sliced layers. Bridges slanting upwards will sag as both edges of the bridge can't be anchored in the same layer.
There are ways to calibrate your printer to handle bridges better, which you can read about at our bridges page. One great test to see how well your printer is with bridges is this Bridge Torture Test.
Add Support Material
Support material can be very useful in printing. In general, though, it is better to avoid it as it can significantly increase your print time and will probably leave some white marks from where it was attached to your model. These break marks may be annoying, but they can be removed in the finishing process.
If you know your model will have issues with overhangs, you can add supports right into your model. To learn how to make effective supports, read our wiki page about designing support structures.
Automatically Generating Supports
Slic3r can automatically generate supports that are specially designed to pull away easily.