Westeros is a feudal kingdom. Which means it consists of a hierarchy of different noble houses serving the one senior to it, and all serving the Crown.
A Great House is one that is answerable only to the Crown. There are no noble houses senior to it. In essence, they are mini kingdoms. In Westeros, at the beginning of the series, the Great Houses are:
- House Stark of the North
- House Baratheon of the Stormlands
- House Lannister of the Westerlands
- House Tully of the Riverlands
- House Arryn of the Vale
- House Greyjoy of the Iron Islands
- House Tyrell of the Reach
- House Martell of Dorne
Under a Great House you have several Noble Houses serving them. They pay taxes, supply military aid in times of need, and submit to the authority of it's Great House. A Noble House is simply a noble family that has, more or less, full control of its territory and commands the allegiance of its inhabitants. Depending on it's size, a Noble House might have several lesser lords serving the more senior lord. For instance, House Royce (a vassal of House Arryn of the Vale) has two branches: House Royce of the Runestone, which is the senior branch, and House Royce of the Gates, which is a less senior cadet offshoot of the main branch. Sometimes, enough time passes that the cadet branch becomes it's own house. For instance House Karstark started out as a cadet branch of House Stark, known as the Starks of Karhold. With time Karhold Starks became Karstarks.
Under a Noble House you can have several Knightly Houses. Those are Knights who have also been granted land (along with its accompanying peasants) as reward for good service. In essence they are proto-Noble Houses, not big enough to have vassals of its own and do not have the authority to serve justice or arbitrate and must submit to its lord in those cases. For instance, House Clegane is a knightly house that swears fealty to House Lannister.