Learn Icelandic: Ég loka hurðinni.

In Icelandic, some verbs naturally require their object to be in the dative case rather than the accusative, and að loka (to close) is one of them. You simply have to memorize that loka always takes a dative object.

How is the word hurðinni built?

It is made from the feminine noun hurð (door) plus the definite article ("the"). Because the verb forces the word into the dative case, we have to use the dative feminine suffix for "the", which is -inni. So, hurð + -inni = hurðinni.

I've also seen the word dyr for "door". Why is hurð used here instead?

Both words mean "door", but they refer to different things! Hurð refers to the physical object (the wooden or metal panel swinging on hinges), while dyr refers to the empty doorway or opening itself. When you say Ég loka hurðinni, you are specifically talking about moving that physical object to shut the space.