Leigutíminn byrjar í dag.

Because Icelandic very often makes compound nouns as single words.

Leigutíminn = leigu + tíminn

• leiga = rent, rental
• leigu- = rental-, rent-
• tími = time, period
• -inn = the

So leigutíminn means the rental period or the lease period. Writing it as one word is completely normal in Icelandic.

The ending -inn is the definite article, meaning the.

In Icelandic, the is usually attached to the end of the noun instead of being a separate word.

• leigutími = rental period
• leigutíminn = the rental period

So where English says the rental period, Icelandic says leigutíminn.

Here it is singular, nominative, masculine, definite.

The basic dictionary form is:

• leigutími = rental period

In the sentence Leigutíminn byrjar í dag, it is the subject of the sentence, so it appears in the nominative case.

That is a very common pattern in Icelandic compounds. The first part of a compound often appears in a modified form, sometimes related to an old case form.

So:

• leiga = rent
• leigu- = rental-, rent-

You do not need to translate this piece by piece every time. It is best to learn leigutími as a normal compound meaning rental period.

Byrjar is the present tense of the verb byrja, which means to begin or to start.

So:

• byrja = to start, to begin
• byrjar = starts / is starting

In this sentence, Leigutíminn byrjar í dag, it means starts today.

Because byrjar is the form that agrees with a third-person singular subject.

The subject is Leigutíminn = the rental period, which is singular.

Compare:

• ég byrja = I start
• þú byrjar = you start
• hann / hún / það byrjar = he / she / it starts

Since leigutíminn is a singular thing, Icelandic uses byrjar.

It can often cover both ideas, depending on context.

Icelandic uses the simple present more broadly than English sometimes does. So Leigutíminn byrjar í dag can naturally correspond to:

• The rental period starts today
• The rental period is starting today

In most learning materials, starts today is the clearest translation.

Í dag means today.

Literally, it is:

• í = in
• dag = day

So word-for-word it looks like in day, but together it simply means today. This is a fixed everyday expression in Icelandic.

Because the preposition í often takes a case other than the basic dictionary form.

The dictionary form is:

• dagur = day

But after í in this expression, it appears as:

• dag

So í dag is the normal idiomatic form for today. This is something learners should memorize as a set phrase.

Yes. Icelandic word order is fairly flexible, especially when you want to emphasize time or place.

These are both natural:

• Leigutíminn byrjar í dag.
• Í dag byrjar leigutíminn.

The second version puts more focus on today. English can do something similar with Today, the rental period starts.

You would remove the definite ending:

• Leigutími byrjar í dag.

That means A rental period starts today or Rental period starts today, depending on context.

But in real usage, the definite form leigutíminn is often more natural if you are talking about a specific rental period already known from context.

A rough guide is:

• Leigutíminn ≈ LAY-guh-tee-minn
• byrjar ≈ BIR-yar or BIR-jar
• í dag ≈ ee daughg

A few helpful points:

• ei in Icelandic often sounds like ay
• í is like a long ee
• j is pronounced like English y

Exact pronunciation depends on accent and speech speed, but that rough guide will help you get close.