This is a clause having a subject and a verb that relates to something else in the sentence.

The type of clause described in the question is a Relative Clause.

Explanation:

A relative clause is a group of words having a subject and a verb that relates to something else in a sentence by describing, adding further information, or modifying that "something else." Thus, a relative clause functions as an adjective, that is, it modifies a noun. This type of clause should always be part of a sentence because it does not express a complete thought on its own.

Here's an example of a sentence with a relative clause (underlined):

The board games that I borrowed are not in excellent condition.

"that I borrowed" is a relative clause because it modifies and relates to the noun "the board games" and it has a subject, which it's the relative pronoun "that", and a verb, which it's "borrowed."