The humble kitchen table is a common piece of furniture found in many small to medium sized US homes, occupying a niche falling between the dining room table and the breakfast table. Any, all, or just one of these various tables may be found in any given house. I myself have a dining room table and a breakfast table, but no kitchen table. My parents have a dining room table and a kitchen table, but no breakfast table.
The differences in the tables can be subtle. Any of the tables may have a removable leaf or two that can be inserted to lengthen it to seat more people when required, a feature not typically found on non-food related tables.
A dining room table is usually a nice, hardwood table set in a separate room of the house designated for dining, seating six or more people, and used primarily when company is visiting. A breakfast table is a small, usually round table in a set-off section from a kitchen referred to as a breakfast nook seating 2-4 people and rarely used for visitors. While I'm on the subject, a coffee table is a short table generally placed in front of a sofa or loveseat to hold beverages and snacks while entertaining guests or just watching television.
A kitchen table is only found in larger kitchens that double as a dining area: a general-purpose, not-too-fancy table appropriate for all meals and most occasions.
In most homes, the dining room table is too nice to use for anything other than eating, so the kitchen table, if present, will typically be used for numerous other activities. Unless a given house also has special-purpose tables, it's not uncommon for a kitchen table to be a board and card game table, a crafts table, a workbench, a homework table, and a play surface for children and their toys.
The Debutante tells me that a breakfast bar with stools is much more common in the UK than a breakfast table is.