Learn Russian: Я часто дарю сестре подарок.

This is the Dative case, which we use for the receiver of an action (the indirect object). The dictionary form is сестра. For feminine nouns ending in -а, we drop the -а and add -е to make them Dative.

Does дарю just mean "give"?

дарю (from the verb дарить) specifically means "to give as a gift." If you were just handing someone a pencil or a book to borrow, you would use a different verb (давать).

The translation says "give my sister a gift." Where is the word for "my" in Russian?

Russian often drops possessive pronouns like "my" or "your" when it's obvious from the context. Because you are the one doing the action (Я), it is automatically assumed you mean your own sister unless you specify otherwise.

подарок (gift) is the object being given. Why didn't its ending change?

Since подарок is the direct object (the thing being given), it is in the Accusative case. For masculine inanimate objects (things that aren't alive), the Accusative form looks exactly like the dictionary form. So, no ending change is needed!