A determiner is a word that introduces a noun, such as a/an, the, every, this, those, or many (as in a cat, the cat, this cat, those cats, every cat, many cats).
The determiner the is sometimes known as the definite article and the determiner a (or an) as the indefinite article.
Words like my, our, your, his, her, its, and their are known as possessive determiners. They come before nouns and indicate ownership of the noun in question, as their name suggests:
My leg hurts.
James sold his business.
Bring your children with you.