Hebrew word from the root sh-m-r meaning "To Watch" or "To Guard". The word can be used in three main contexts.
Be it on an army base, or keeping watch over a dead body before the funeral, a Shomer is someone who does Shmirah - keeping a watch out.
Common terms are "Shomer Shabbat" or "Shomer Kashrut". These can literally be translated as "Guarding Shabbat" and "Guarding Kashrut". The meaning is someone who keeps the laws of Shabbat or Kashrut.
This is the more common usage nowdays. A Shomer is someone who watches the preparation of food in a Kosher restaurant. Why is this needed?
The Jewish dietary laws are complex. Even most Orthodox Jews don't know all of them. And a lot of the chefs in Kosher restaurants aren't actually Jewish themselves. The job of the Shomer is to keep an eye on all of the food coming in (to ensure it is from Kosher sources), and to watch all of the preparation to ensure that only methods that are within the Jewish law are used.
If a Shomer was to rule a batch of food not Kosher, it has to be disposed of (or perhaps given to a non-Jewish charity). One could say that this discourages the Shomer from doing this, for fear of losing his salary or his job. For this reason, a Shomer must not be directly employed by the restaurant. Instead, they are usually employed by the Beth Din, and therefore they have no personal loss if they feel that, for reasons of Kashrut, food must be ruled not Kosher. The restaurant then pays the Beth Din to have a Shomer on the premises, and the Beth Din issues a certificate (usually annually, although sometimes quarterly) saying that the restaurant is under its supervision.
Some people have said to me "This sounds like a form of religious police". This couldn't be further from the truth. A restaurant owner is never forced to be supervised for Kashrut. But look at it this way.
I only eat Kosher food. I know my kitchen at home is Kosher. I also know my parents have a Kosher kitchen. And I know that most of my friends do, so I'll eat there. But who knows what happens in a restaurant. I don't know the owner personally. Even if I did, I don't know all the chefs personally. And they may not even be Jewish. Furthermore, they may be concerned about their profits, and therefore less likely to rule a borderline case as not Kosher than I'd like. By only eating at a restaurant with an appropriate certificate, I know that the food is 100% Kosher.