Quick answer
The Norwegian alphabet has 29 letters: A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z, Æ, Ø, Å. The first 26 are identical to English; the final three — Æ, Ø, Å — are extra vowels unique to Norwegian and Danish.
Æ, Ø, Å — The Three Extra Letters
These three vowels are what make Norwegian look distinct. They are full letters, not accents, and they change the meaning of a word.
Æ æÆ — the open vowel
Pronounced like the "a" in English "cat" or "bad".
Æ is a ligature of A and E — literally "ae" written as one letter. It dates back to Old English and Old Norse, and survived in Norwegian while English dropped it in the Middle Ages. You will meet it in words like ære (honour), lære (to learn), være (to be), and the iconic single-letter word æ — a dialect form of "I" used in Trøndelag. Æ is the rarest of the three extras but appears often enough that you cannot avoid it. To type it on macOS: Option + '.
Ø øØ — the rounded front vowel
Similar to the German ö or the "i" in "bird" (British English).
Ø is a stylised O with a slash — historically an O and E merged together. To make the sound, shape your mouth for an "ee" but round your lips as if saying "oo". Common words: øy (island), øl (beer), smør (butter), søster (sister), første (first). Ø also shows up in many place names — Tromsø, Bodø, Drøbak — so you will see it on maps and road signs constantly. Swedish writes the same sound as Ö. To type it on macOS: Option + O.
Å åÅ — the long "aw"
Pronounced like the "o" in "more" or "aw" in "saw".
Å is the newest of the three — officially added to Norwegian in 1917, replacing the old double-a spelling "aa". You will still see the old form in some names (Haakon, Aalesund), but modern Norwegian uses Å everywhere else. Core words: år (year), så (so/then), gå (to walk), få (to get), blå (blue), and the tiny fishing village literally named Å in Lofoten. Å shares the same letter and sound as Swedish Å. To type it on macOS: Option + A.