Learn Icelandic: Innritunarborðið er við innganginn. - The check-in desk is by the entrance.

While many prepositions describing a static location (like í or á) take the dative case, við behaves differently. When við means "by," "next to," or "at" in a spatial sense of proximity, it takes the accusative case.

Could I use hjá instead of við to mean "by the entrance"?

Yes, you could say hjá innganginum (note that hjá always takes the dative case). However, við often implies being right up against or functionally attached to a space. Since a check-in desk is usually stationed exactly at the entrance, við is a very precise and natural choice here.

What is the dictionary form of innganginn and how did it change here?

The base noun is the masculine word inngangur (entrance). To make it accusative, the -ur ending drops, leaving inngang. Then, we add the masculine accusative definite article -inn to get innganginn (the entrance).

How is the compound word innritunarborðið formed?

It is made of innritun (check-in or registration) and borðið (the desk or table). The -ar in the middle is the genitive (possessive) ending for innritun, essentially making this "the desk of check-in."