Learn Danish: Hun skrev et digt til sin mor. - She wrote a poem for her mother.

The word sin is a reflexive possessive pronoun. It is used because the mother belongs to the subject of the sentence (hun). If you used hendes here, it would mean she wrote a poem for another woman's mother. Because mor is a common gender noun (an en-word), we use sin rather than sit or sine.

The English translation says "for her mother", so why use til instead of for?

In Danish, when you give, send, or make something directed to a recipient, you generally use til. It acts more like the English "to" (she wrote a poem to her mother). If you used for, it would usually imply she did it on her mother's behalf, for example, if the mother had a writing assignment and the daughter did the work for her.

What is the infinitive form of skrev?

It is the past tense of the irregular (strong) verb at skrive (to write). As a B1 learner, it is a good idea to memorize its primary forms: skriver in the present, skrev in the past, and har skrevet in the present perfect.

Is the word for poem, digt, an en-word or a t-word?

It is a neuter noun, or a t-word, which is why it uses the indefinite article et. If you wanted to talk about "the poem" instead of "a poem" in this sentence, you would attach the definite suffix to make it digtet.