Never buy a house if you can buy a house and its appurtenances. Appurtenance is legalese for "a thing connected to something else." How's that for precision? In fact, you could substitute the word "thing" for appurtenance in most sentences without any loss of meaning. In the context of that house purchase, an "appurtenance" would be things like fixtures that go along with the house.
The word comes from Anglo-French apurtenance, and ultimately from the Latin appertinire ("to belong"). As the Latin root suggests, an appurtenance was originally a right that belonged to some other property or position. Hence the phrase "the rights and privileges appurtenant thereto." We're all connected, but we're not all appurtenant.