That is a chemical called hydrogen sulfide. It's not harmful, but even a small amount can really stink.
Hydrogen sulfide occurs naturally and can build up in your appliances or pipes over time until you begin to smell it. There are three places that are usually the origin of the rotten egg smell. To find which one it is will require a little aquatic detective work. The first step is to run hot and cold water in all the faucets of your house at the same time. Quickly go from room to room and give each fixture a sniff. Smell them together, then turn off one at a time to get an individual whiff. Note whether the smell is coming from all faucets or just those in one particular room.
If the smell is coming from the hot water only, it's your water heater.
If the smell is only coming from one room, it's likely a problem with your pipes.
If the smell is everywhere, it's coming from your plumbing system or groundwater.
Now that you have a better idea of the source, here are some possible solutions. If you suspect the issue is with your water heater, disinfect it by turning up the temperature on it to 160 degrees for a couple of hours to kill any bacteria. If that doesn't do the trick, you should consider calling a professional plumber to hone in on the source of the problem.
If the smell is coming from just one room, run the faucets in that room for at least ten minutes to try and clear the pipes. If the smell is in your whole house, try the same thing by running all the faucets at once for about ten minutes.
Water Softeners do not remove odors from water. Modern filtration systems only need to changed once per year and can part of a plumbing maintenance agreement.