Clauses that cannot stand alone and are not complete thoughts are called

These are known as dependent clauses.
 Independent clauses are clauses that can stand alone in sentences if the sentence was chopped up. However, dependent clauses cannot do that.
For example: "When I ran outside, the dog was jumping."
The dependent clause is "When I ran outside," as it cannot stand alone and must have another clause  to be completed.
The independent clause is "the dog was jumping.", as the sentence can be put by itself and does not need another clause to be completed.