المستأجر يوقع عقد الإيجار.

Because عقد الإيجار is an iḍāfa construction, often called a genitive construct.

In an iḍāfa:

• the first noun does not take ال
• the second noun determines whether the whole phrase is definite or indefinite

So:

• عقد إيجار = a rental contract / a lease contract
• عقد الإيجار = the rental contract / the lease contract

Even though عقد has no ال, the whole phrase is definite because الإيجار is definite.

It is an iḍāfa phrase: noun + noun.

Here:

• عقد = contract
• الإيجار = the renting / the lease / the rental

Together, the phrase means something like:

• lease contract
• rental contract
• contract of lease

In Arabic, this structure is very common. English often uses of or a noun used like an adjective, but Arabic usually uses iḍāfa instead.

Here, المستأجر means the tenant or the renter/lessee.

A very useful point: in unvocalized Arabic script, المستأجر can look ambiguous. With vowels:

• المُسْتَأْجِر = the tenant / the one who rents
• المُسْتَأْجَر = the rented one / leased

So the exact reading depends on context. In this sentence, it clearly means the tenant, because the tenant is the one signing the lease.

Also compare:

• مؤجِّر = lessor / landlord / one who rents out
• مستأجِر = lessee / tenant / one who rents

Both are possible in Arabic.

• المستأجر يوقع عقد الإيجار
  starts with the noun, so it feels more like The tenant signs the lease contract with focus on the tenant

• يوقع المستأجر عقد الإيجار
  starts with the verb, which is also very natural in Arabic

So Arabic allows both subject-first and verb-first word order.

Very roughly:

• subject first can sound a bit more topic-focused
• verb first is often a more neutral narrative order

يوقع is the present/imperfect form of وقّع.

• past: وقّع = he signed
• present/imperfect: يوقّع = he signs / he is signing

This verb is commonly used for signing documents:

• وقّع العقد = he signed the contract
• يوقّع العقد = he signs / is signing the contract

So in this sentence, يوقع عقد الإيجار means signs the lease contract.

It can mean either, depending on context.

The Arabic imperfect tense often covers both:

• simple present: signs
• present progressive: is signing
• sometimes even a near future sense, depending on context

So:

• المستأجر يوقع عقد الإيجار could mean
  The tenant signs the lease contract
  or
  The tenant is signing the lease contract

Context tells you which is intended.

In fully vocalized MSA, the sentence is:

الْمُسْتَأْجِرُ يُوَقِّعُ عَقْدَ الإِيجَارِ

The endings are:

• الْمُسْتَأْجِرُ: nominative, because it is the subject
• يُوَقِّعُ: the verb in the indicative
• عَقْدَ: accusative, because it is the direct object
• الإِيجَارِ: genitive, because it is the second part of the iḍāfa

So the grammar is:

• subject = nominative
• object = accusative
• second noun in iḍāfa = genitive

A careful MSA pronunciation is:

al-mustaʾjiru yuwaqqiʿu ʿaqda al-ījāri

If you are reading without case endings, you will often hear something closer to:

al-mustaʾjir yuwaqqiʿ ʿaqd al-ījār

A few pronunciation notes:

• ع in عقد is the letter ʿayn, which has no exact English equivalent
• ق in يوقع and عقد is qāf, a deeper sound than English k
• the doubled consonant in يوقّع should be held slightly longer

Because Arabic has no separate indefinite article like English a/an.

Arabic usually shows indefiniteness by:

• absence of ال
• often tanwīn in fully vocalized text

For example:

• مستأجرٌ = a tenant
• عقدُ إيجارٍ or عقدَ إيجارٍ = a lease contract, depending on grammar in the sentence

In your sentence, المستأجر is definite because of ال, and عقد الإيجار is definite because it is an iḍāfa whose second noun is definite.

A natural MSA version would be:

يوقعُ مستأجرٌ عقدَ إيجارٍ

or, with subject first:

مستأجرٌ يوقعُ عقدَ إيجارٍ

Here:

• مستأجرٌ = a tenant
• عقدَ إيجارٍ = a lease contract

Notice the difference from your original sentence:

• المستأجر = the tenant
• عقد الإيجار = the lease contract

So Arabic marks definiteness very clearly.

It overlaps with English rent, rental, and lease, depending on context.

الإيجار can refer to:

• the act of renting/leasing
• rental/lease as a legal arrangement
• sometimes the rent itself, depending on context

In عقد الإيجار, the most natural English translation is usually:

• lease contract
• rental contract
• lease agreement

So it is broader than a single English word, and the best translation depends on context.