Learn Polish: Wyślę wnukowi paczkę, bo ma dzisiaj urodziny. - I will send my grandson a package, because it's his birthday today.

Wnuk is a masculine noun meaning "grandson". Because he is the recipient of the package (the indirect object), he needs to be in the dative case. For most masculine nouns, you form the dative singular by adding -owi, which gives us wnukowi.

The English translation says "my grandson", but where is the word for "my"?

In Polish, possessive pronouns like "my" (mój) are often dropped when the context makes it obvious. Since you are the one doing the sending, it is automatically assumed it is your grandson. You could explicitly say mojemu wnukowi, but it is not necessary here.

Wyślę looks like a present tense verb. How do we know it means "I will send"?

Wyślę comes from the perfective verb wysłać (to send). Perfective verbs focus on the completion of an action, so they do not have a true present tense. When you conjugate them using present tense endings, they automatically express the future tense!

Why does it say ma dzisiaj urodziny instead of "it is his birthday"?

In Polish, you do not say that "it is" someone's birthday. Instead, a person "has" a birthday. The verb ma means "he has". Also, notice that the word for birthday, urodziny, is a plural noun in Polish, so you are literally saying "he has birthdays today".