A restaurant which looks as if it could be / should be Kosher but isn't.
Most Kosher restaurants around the world are very obviously Jewish / Kosher. They have Hebrew writing on the walls and menu, they serve very Israeli or Jewish style food etc.
Then there are stealth-kosher restaurants, which are strictly Kosher, but come across as a typical decent restaurant, serving "normal" food. You wouldn't know it's Kosher except for a small certificate on the wall somewhere, and the fact that there's no pork on the menu.
Kosher Style restaurants are the opposite of this. They are restaurants which look Kosher, serving "Jewish" food (e.g. Chicken Soup) but aren't supervised as strictly Kosher, and with no guarantee to the source of the ingredients. Orthodox Jews won't eat at these restaurants, but they are very common in places with large numbers of non-Orthodox Jews (eg New York). If you are concerned about Kashrut, always check for a certificate from a recognised authority before eating at a restaurant. Most owners of Kosher Style restaurants aren't out to make life hard for religious Jews, but on the other hand, they're not going to be shouting out to turn away business.
An example. Katz's Deli in New York is Kosher Style, but not Kosher (although apparently it used to be). A lot of the food they serve there may be Kosher, but there is no supervision to check they are keeping all of the laws of Kashrut in its preparation. On the other hand, the Second Avenue Deli (not so far away from it) is supervised Kosher.