Flux is a chemical that we use to control oxidation during soldering. It's a necessary part of the soldering process. However, at the end of the soldering you generally are left with a residue that covers your circuit board, and with enough time the solder flux may actually begin to corrode the parts of your circuit. It's a process that may take years and take a long time to occur, but it's something that you may want to control.
In order to do so, the standard rosin flux is rather resistant to most kinds of solvents, and so we have to use a particularly toxic solvent known as trichloroethylene. This is also known as dry cleaning fluid and is what's responsible for giving your dry cleaned clothes that little bit of smell when you get them home from the dry cleaners. This is a solvent that's effective at dissolving the flux.
In order to avoid the use of this solvent you can either leave the flux residue on the board undisturbed and know that your circuit may deteriorate over the years, or you may look for an alternative flux formulation. There's water soluble flux which can be cleaned with water and detergent. There are mild fluxes that can be left on the board. And there are no clean fluxes that leave virtually no residue at all. These can be selected instead of the rosin flux and be just as effective.
If you do decide to go ahead and use rosin flux, and trichloroethylene to clean, you'll want to take precautions against the vapors. First, you'll want to protect your skin using gloves and perhaps a coverall to make sure that it doesn't get onto your hands or your skin. You'll also want to make sure that the area is well ventilated, because the vapors will build up very quickly.
In states like California the trichloroethylene, I believe, is banned. You're not legally allowed to use this in that state. It's not something you may want to use in widespread use. Make sure to consider your state laws and make sure that it's legal to use where you're planning on using it.