Dyes
Ethidium bromide is the most common dye for staining DNA electrophoresis gels. It inserts itself between the two strands of the DNA molecules in your sample. Unfortunately, it can slip between strands not only in your sample but in your cells as well, so it has to be handled carefully; wearing double gloves is a customary precaution. Recent years have seen the introduction of alternative chemicals like SYBR Safe as well.
Fluorescence
Once you've added your dye, it will bind to the DNA in your sample, causing it to become highly fluorescent. Place the gel under a black light and the DNA bands will glow brightly, standing out in vivid contrast to the background. The dye makes it possible to see the DNA bands on the gel, which would otherwise be invisible to you. It's essential you refrain from looking at or into the black light while viewing the gel, because the UV could badly damage your eyes.
Analysis
Lab technicians typically take pictures of the gel with a digital camera; at that point they can discard the gel and save the picture on a hard drive to review it later. You can glean a lot of information from the position and types of bands you see. If you ran size markers on the gel in addition to the sample, for instance, you can get a rough estimate of the size of your DNA molecules and the number of base pairs they contain.
Other Features
If you used a special type of protein to cut a piece of DNA, you could run it on a gel and check for multiple bands in the same lane to make sure the restriction enzyme worked. Alternatively, if you're working with small loops of DNA called plasmids, and you want to make sure a plasmid contains a segment you inserted into it, you could cut it with a special protein and run it on a gel to make sure the segment was present.