What type of clause has a subject and verb but does not express A complete thought?

A clause that has a subject and a verb but does not express a complete thought is a dependent clause, also called a subordinate clause.

Since traditionally a complete independent sentence is said to "express a complete thought", an example of a clause which cannot stand alone as a complete sentence would be one such type. An independent sentence in English must have a tensed verb, so I guess we are looking for a clause whose verb is not tensed. An infinitival clause with "to" would be one such example. In "It would be too difficult for us to clean all these fish", for instance, "for us to clean all these fish" is a clause with subject "us" and verb "clean", but there is no past or present tensed form of a verb, so it cannot stand alone as a complete sentence.

Other similar examples use an "-ing" form of a verb, rather than the "to" infinitive form.