Learn Icelandic: Ég opna hurðina.

The dictionary word for "door" is hurð. Because you are saying "the door" and it is the direct object of the verb (the thing being opened), it takes the accusative definite ending. For feminine words like hurð, that ending is -ina, giving us hurðina.

Why does hurð take the accusative case here?

The verb opna means "to open". The thing that you open is the direct object of the sentence because it is receiving the action. In Icelandic, direct objects are usually put into the accusative case.

Wait, isn't opna the dictionary form of the verb?

Yes, the infinitive (dictionary form) for "to open" is að opna. However, the "I" form (first-person singular) in the present tense is exactly the same: ég opna.

How do I know the gender of hurð based on this sentence?

The ending -ina tells you that it is a feminine noun! When a feminine noun is in the accusative case and has "the" attached to it, it gets the -ina ending. If it were masculine, it would usually end in -inn, and if it were neuter, it would end in -ið.