Colloidal silver is a suspension of very small silver particles in water (the term “colloid” refers to a mixture in which one substance is dispersed evenly throughout another (source)). Colloidal silver is a very intensely-discussed topic in some health circles, with passionate supporters on both sides; some who claim it is a wonderful cure-all, others who say it offers no benefit but may be risky.
Silver is claimed to have anti-bacterial and anti-viral properties (it is sometimes used in air filters and to treat burns), and the theory is that, when consumed, the tiny particles found in the colloidal suspension can attack whatever bacteria or virus may be attacking the body.
Some people drink the mixture, letting their blood take it where it is needed, and other people gargle with it so it can directly attack the bacteria or virus in the sore throat.
There are some people online who make great claims about the efficiency of colloidal silver, so as always, do your own due diligence. There are some side effects reported which include killing of the beneficial bacteria in the intestines (if you drink it) and Argyria, a condition where the skin turns blueish/gray, although that is thought to only happen with high doses used for long periods of time, and sometimes with improperly made mixtures.
Another thing to keep in mind is that there are no laws about the production process, so just because you buy something called “colloidal silver” doesn’t mean it was made properly, or at the supposed dosage or strength.