Those were the years that I remember best.  The bolded words are what kind clauses?

The bolded words refer to an adjective clause."that I remember best" is acting as adjective, used to describe "the years."generally, adjective clauses will start with a relative pronoun, making them easier to identify. "that" is a relative pronoun.

It's an adjective because it is a description. The thing it's describing are the years. So the "that I remember best" are describing the significance of the years.