Where: France, Paris in particular.

When: Shortly after the Revolution, often considered a direct result of the French Revolution. French chefs were already known to be excellent around Europe and with the flight and execution of the upper strata of French society many chefs who had previously found employment as chefs to wealthy households were suddenly without a job, this led to the creation of restaurants. The new revolutionary government's elimination of the guilds which had previously held control of such trades also played a part.

What were they like? They catered to the remaining wealthy and newly developing bourgeoisie class, we know there was a great emphasis on sauces and patisserie usually meats covered in sauces and sweet delicate baked constructions (pièces montées). Here is a drawing of a restaurant "in action" from the period:

https://www.mtholyoke.edu/courses/rschwart/hist255-s01/pleasure/restaurants_revolution.html

Edit: As noted below there are other eating houses made mention of pre revoltion in Paris and London, some of which were hotels etc. which provided food and became known for it, whether some of these places could be classified as restaurants is hard to say and we know very little of their operation.