Learn German: Das Haus gefällt der Frau. - The woman likes the house.

Normally, yes! In the basic nominative (subject) case, der is masculine and die is feminine. But here, the woman is in the dative case. In the dative case, the feminine article die changes to der. It takes a little getting used to, but in this sentence, der Frau means "to the woman".

Why is the woman in the dative case here anyway?

It is because of the verb gefallen (to like / to be pleasing). In German, this verb works backward compared to English. You don't literally say "The woman likes the house." Instead, the grammar translates closer to "The house is pleasing to the woman." The person who receives that feeling takes the dative case.

If the grammar is backward, does that mean Das Haus is the subject?

Exactly! Since the structure is literally "The house is pleasing to the woman", Das Haus is the subject of the sentence (in the nominative case). The verb must match this subject, which is why we conjugate it for "the house" (it) rather than "the woman" (she).

Why does the verb gefällt have an umlaut (ä)?