Learn Icelandic: Kötturinn hleypur fyrir bílinn. - The cat runs in front of the car.

Because the cat is moving to a new position. The preposition fyrir takes the accusative case when it shows direction or movement to a destination. If the cat were just sitting stationary in front of the car, fyrir would take the dative case instead (fyrir bílnum).

I thought fyrir meant 'for'. Why does it mean 'in front of' here?

Fyrir is a very flexible preposition! While it often means 'for', when talking about physical space and movement, it translates to 'in front of'. The context of a cat running tells us it is about location.

Why does bílinn only have one 'l' in the middle, when the basic word is bíll?

Masculine words ending in -ll like bíll usually drop one l in the accusative case before adding the definite article. So it becomes bíl plus -inn, making bílinn.

The verb hleypur looks very different from its dictionary form. What is the infinitive?

The dictionary form is hlaupa (to run). It is a strong verb that has a vowel shift in the present tense singular, changing the au to ey. This is why 'it runs' becomes hleypur.