There are a few types of breakfast eating surfaces that use breakfast table stools. Of course, because they are so much higher than regular chairs, you cannot use stools with any breakfast table. The first kind of breakfast table that uses stools is what is called a high top table. High tops sit at chest level rather than thigh level. While it is not a requirement, you will find that most high tops are meant for two or three people. It is very unusual to find a high top table for six or larger. One attribute that many high top tables have that is great for people sitting on stools is a bar or piece that acts as a foot rest. It can be uncomfortable to sit on a stool and have your feet dangling. This foot rest makes it much more comfortable to sit on stools and it encourages good posture. In order to reach the foot rest you need to sit closer to the front of the stool and lean slightly forward which encourages good posture.
The other type of breakfast table with stools is not a table at all. Instead, it is your kitchen island. Islands are great because they give the cook extra cooking and preparation surface. But there is an added benefit in that you can use the surface as a kitchen table. If you do not have a separate breakfast nook then kitchen island table seating is a great way to have the breakfast nook feel. If you plan to use your kitchen island as a breakfast table and you have the opportunity to design or choose it then make sure that the counter of the island is larger than the cabinets underneath so that there is room for your knees and legs when you are sitting at the breakfast island. One great variation on the normal kitchen island is one where you have one surface for cooking and food preparation and then a section around one end that is higher for eating.
While these are the most common types of tables that use stools, you can find low stools that are at the same height as regular chairs that can be used at any breakfast table.