What is the difference between counter and table, in the context of a surface that one might eat at (as in the kitchen counter or the kitchen table)?
Counter can be used as short for countertop, which is (NOAD) a flat surface for working on, especially in a kitchen. The first definition given from the NOAD for counter is
I would interpret counter as to mean, for example, the fixture you can usually see in an American kitchen and that is used for breakfast; its structure reminds me of the counter you find in a bar. The difference with the table is that the table is less height.
When I visit USA in my frequent trips, I hear the word counter being used as synonym of countertop, which is a flat surface for working on, especially in a kitchen. This is could be valid in the specific place I go (Long Island, Suffolk County), though.
The counter is a built-in fixture, typically consisting of the countertop (made of stone, sturdy plastic, Formica, or concrete) and the cabinet (made of sheet-metal or wood). In the US, the surface of the counter is almost universally 36" off the floor. It's usually made to measure.
A table is a piece of furniture, generally made entirely of wood (although metal-and-Formica creations are not unknown) and consisting of the tabletop and either legs or a central pedestal. The top surface is about 30" from the floor (comfortable for eating but too low for working from a standing position). A table is generally purchased ready-made from a store that also sells chairs, couches, and other furniture.