When you go to a bar what's it mean when they say single or double?

Single or Double Shot.

When they ask single or double they are generally referring to the shot. The reason that there are different size glasses is that different types of drinks come in different sizes and it is logical to assume that different people order different drinks, not everyone there would order the exact drink you did.

Yea a double would just be a double drink to prevent from having to order as often.

What kind of bar were you in where the customers were drinking from cups? I thought that went out when Prohibition ended.

single shot or double