What type of pronoun is that in the sentence That is my cousin?

The pronoun 'that' in the example sentence is a demonstrative pronoun.

A demonstrative pronoun takes the place of a noun (cousin) indicating near or far in place or time.

The demonstrative pronouns are: this, that, these, those.

Note: The demonstrative pronouns are adjectives when placed before a noun to describe that noun (that cousin, that noun).