Filaments are large regions of very dense, cool gas, held in place by magnetic fields. They usually appear long and thin above the chromosphere, as in the picture above. It is because they are cooler than their surroundings that they appear dark. But if they appear on the "edge" of the Sun, they appear brighter than the dark outer space behind them. In that case we call them prominences. Here is a close-up picture of a prominence.
Prominences and filaments are really the same thing, but they look bright or dark depending on what is in the picture's background. At the right is another picture that shows a bunch of things. This is another H-alpha picture, and you can see the dark thread-like filaments on the Sun. You can also see some bright-looking prominences sticking out beyond the "edge" of the Sun on the far right. In the box, you can see one object that appears to be both filament and prominence! The part on the face of the Sun is dark (a filament) and the part hanging out past the edge is brighter than the empty space behind it (a prominence). You can see that it's all one piece, the only difference is how the object looks compared to what's behind it in the picture.
Filaments last for a few weeks or months. The gas in a filament will eventually move to a different layer in the Sun and will no longer be visible in an image of the chromosphere. But at the same time, other gas may move into the chromosphere and create a new filament someplace else. The birth and death of filaments is a mystery and the subject of ongoing study by solar scientists.
Image Credits: Sun in Hydrogen-alpha: NOAA/SEL/USAF
Prominence close-up: Extreme Ultraviolet Telescope, Solar and Heliospheric Observatory
Disk/limb composite: Mauna Loa Solar Observatory