Skráningarnúmerið er í ökutækjaskránni.

A natural breakdown is:

• skráningarnúmerið = the registration number
• er = is
• í = in
• ökutækjaskránni = the vehicle registry / the vehicle register

So grammatically:

• skráningarnúmerið is the subject
• er is the verb
• í ökutækjaskránni is a prepositional phrase telling you where it is

That -ið is the definite article attached to the noun. Icelandic usually puts the at the end of the word instead of using a separate word like English the.

So:

• skráningarnúmer = a registration number / registration number
• skráningarnúmerið = the registration number

This noun is neuter singular, and -ið is a common definite ending for neuter nouns in the nominative singular.

Because it is:

1. definite = the vehicle registry
2. in the dative singular
3. a feminine noun

The base noun is ökutækjaskrá.

Its forms here are:

• ökutækjaskrá = a vehicle registry
• ökutækjaskránni = in the vehicle registry / the vehicle registry in the dative definite form

The -nni ending is what you expect for many feminine nouns in the dative singular definite.

Because the preposition í can take either:

• dative for location: in, inside, at rest
• accusative for motion into

Here the sentence describes location, not movement:

• The registration number is in the vehicle registry

So Icelandic uses í + dative:

• í ökutækjaskránni

A very common Icelandic pattern is:

• í + dative = being in something
• í + accusative = going into something

Compare the idea:

• Það er í skápnum = It is in the cupboard
  
  • location, so dative
• Ég set það í skápinn = I put it into the cupboard
  
  • motion/direction, so accusative

In your sentence, nothing is moving, so dative is used.

It is in the nominative singular because it is the subject of the sentence.

In a sentence with er (is), the thing that is something or is somewhere is normally in the nominative:

• Skráningarnúmerið = the subject
• er = is
• í ökutækjaskránni = where it is

So this is a very standard Icelandic sentence structure.

It is a compound noun, which is extremely common in Icelandic.

It breaks down like this:

• ökutæki = vehicle
• ökutækja- = a linking form based on the genitive plural: of vehicles
• skrá = register, registry, record

So ökutækjaskrá literally means something like:

• registry of vehicles
• or more naturally, vehicle registry

This use of a genitive form inside compounds is very common in Icelandic.

Not exactly.

• skráningarnúmer = registration number
• a license plate is the physical plate on the vehicle

In everyday use, people may connect the two very closely because the registration number appears on the plate, but the words are not strictly identical.

So if you want to be precise:

• skráningarnúmer = the number
• the plate itself is something else

You can, but it changes the meaning.

• Skráningarnúmerið er í ökutækjaskránni = The registration number is in the vehicle registry
• Skráningarnúmer er í ökutækjaskrá = A registration number is in a vehicle registry or Registration number is in vehicle registry, which sounds less natural without context

In your original sentence, both nouns are definite:

• the registration number
• the vehicle registry

So the endings matter a lot.

A rough English-friendly guide would be:

SKRAU-ning-ar-NOO-meh-rith er ee UH-kuh-tye-kya-skroun-nih

A few notes:

• á is pronounced roughly like ow in cow
• ú is like oo
• æ is like eye
• ð is a soft sound, often like th in this, though in some positions it is very light
• ll, nn, rn, and other clusters can sound different from what an English speaker expects, so listening to native audio helps a lot

The stress in Icelandic is usually on the first syllable of each word:

• SKRÁningarnúmerið
• ER
• Í
• Ökutækjaskránni

Yes, but Icelandic has an important verb-second tendency in main clauses. Your sentence uses the most straightforward order:

• Skráningarnúmerið er í ökutækjaskránni

You could also front the prepositional phrase for emphasis:

• Í ökutækjaskránni er skráningarnúmerið

That can sound more marked or literary depending on context, but it is grammatical. The key thing is that the finite verb er still comes early, typically in the second position of the clause.

So for learners, the original version is the safest and most neutral word order.