Learn Icelandic: Systir mín vinnur hér.

In Icelandic, possessive pronouns like mín (my) normally come after the noun they belong to. You can put them before the noun if you want to add strong emphasis, but putting them after is the standard, natural way to speak.

The lesson teaches minn for "my", so why is it mín here?

Possessive pronouns must match the gender of the noun they describe. Because systir (sister) is a feminine word, you have to use the feminine form mín. The word minn is the masculine form.

Don't nouns usually need a "the" ending when used with a possessive, like bókin mín (my book)?

Yes, most nouns take the attached "the" ending when used with a possessive pronoun. However, close family members like systir, bróðir, mamma, and pabbi are a special exception. They usually stay in their basic form without the "the" ending.

What is the dictionary form of the verb vinnur?

The dictionary form (infinitive) is vinna (to work). It changes to vinnur here because the subject of the sentence, systir mín (my sister), is a "she" (third-person singular).