Learn Icelandic: Við tökum glasið úr uppþvottavélinni. - We take the glass out of the dishwasher.

The preposition úr means "out of," and it is used when moving something out of a container or an enclosed space—like a dishwasher! If you used frá, it would mean you are moving the glass "away from" the outside of the machine, rather than taking it from the inside.

How does the preposition úr change the word for "the dishwasher"?

úr always forces the following noun into the dative case. The base word is uppþvottavél (a feminine compound of uppþvottur for "washing up" and vél for "machine"). When you add the feminine definite article ("the") in the dative case, it takes the ending -inni, giving us uppþvottavélinni.

Why isn't glasið also in the dative case?

Because glasið (the glass) belongs to the verb taka (to take), not the preposition. It is the direct object you are taking, so it goes in the accusative case. The preposition úr only affects the noun that directly follows it as the starting location.

glasið looks exactly like the dictionary (nominative) form. How do I know it is accusative here?

glas is a neuter noun. A handy rule for neuter nouns is that their nominative and accusative forms are always exactly the same, whether they have the definite article (-ið) attached or not. You know it is accusative here simply because it is the object receiving the action of the verb tökum.