Learn Hardcore Icelandic: Ég þarf leyfi.

Ég ≈ yeh (long vowel; the g is often a soft, breathy sound or silent).
þarf ≈ tharf (þ like English th in thin; a tapped/trilled r; final f like English f).
leyfi ≈ LAY-vi (ey like ay in day; f between vowels sounds like v; final i like the vowel in bit).

Why is there no word for the English a/an?

Icelandic has no indefinite article. A bare noun is indefinite by default. So leyfi can mean a permit or permission in general. To make it definite, you attach a suffix: leyfið = the permit/permission.

What case is leyfi here, and why doesn’t it look different?

þurfa (to need) takes a direct object in the accusative.
leyfi is a neuter -i noun whose nominative and accusative singular look the same: leyfi.
You can see the case on an adjective: Ég þarf nýtt leyfi (I need a new permit; neuter acc/nom singular). In the idiom þurfa á X að halda the noun is dative: Ég þarf á nýju leyfi að halda.

How do I say I need the permission/the permit?