In the example:
Mike found the keys that he had lost.
that is a relative pronoun which introduces the relative clause that he had lost.
In English the relative pronouns are who, whom, which, whose, and that.
who - subject or object pronoun for people
They caught the lady who killed her baby.
I know the man who you met.
which - subject or object pronoun
I read the book which is on the table.
I visited the town which you told me about.
which - referring to a whole sentence
They were unsuccessful which is disappointing.
whom - used for object pronoun for people, especially in non-restrictive relative clauses (in restrictive relative clauses use who)
The boy whom you told me about got the best grades in mathematics.
that - subject or object pronoun for people, animals and things in restrictive relative clauses (who or which are also possible)
I like the vase that is over there.