Eniştem bugün izinli olduğu için yeğenimi parka götürdü.

Eniştem means my brother-in-law.

It is built like this:

• enişte = brother-in-law
• -m = my

So:

• enişte → brother-in-law
• eniştem → my brother-in-law

A useful point: after nouns ending in a vowel, the 1st person singular possessive is often just -m, not -im. So you get:

• araba → arabam = my car
• enişte → eniştem = my brother-in-law

Also, enişte is a fairly specific family term in Turkish, not a catch-all for every kind of brother-in-law.

It can mean either.

Turkish often does not mark gender in family words the way English does. So:

• yeğen = niece / nephew
• yeğenim = my niece / my nephew
• yeğenimi = my niece / my nephew (as the object)

You usually understand which one is meant from context.

Because it is the direct object of the verb götürdü (took), and it is a specific/definite object.

Compare:

• yeğenim = my niece/nephew
• yeğenimi = my niece/nephew as a specific object

Here, my niece/nephew is the person being taken, so Turkish uses the accusative ending.

Breakdown:

• yeğen = niece/nephew
• yeğenim = my niece/nephew
• yeğenimi = my niece/nephew + accusative

This is very common in Turkish: possessed objects are usually specific, so they often take the accusative.

Because parka shows direction toward a place: to the park.

• park = park
• parka = to the park

This is the dative ending: -a / -e.

Compare:

• parka gitmek = to go to the park
• parka götürmek = to take someone to the park

If you said parkı, that would be the accusative, which marks a direct object, not a destination.

Here izinli means something like:

• off
• on leave
• not working that day with permission

So bugün izinli means off today or on leave today.

In many contexts, izinli is used for someone who has official leave from work, school, military service, etc. In this sentence, the natural English meaning is he had the day off today.

Olduğu için is a very common Turkish pattern meaning because he/she/it is/was ... or more literally because of the fact that he/she/it was ...

In this sentence:

• izinli = off / on leave
• olduğu için = because he was

So:

• izinli olduğu için = because he was off / because he was on leave

This structure is extremely common with adjectives and nouns:

• hasta olduğu için = because he/she is-was sick
• meşgul olduğu için = because he/she is-was busy
• evde olduğu için = because he/she is-was at home

You do not need to translate it word-for-word every time; it is often best to just recognize X olduğu için as because he/she was X.

Because this is a subordinate clause, not a standalone sentence.

• oldu = he/she/it became / was
• olduğu = a form used inside a larger structure, such as because he was...

Turkish often uses special clause forms before words like için.

So:

• izinli oldu = he became/was off
• izinli olduğu için = because he was off

You can think of olduğu here as the form Turkish needs in order to attach için and make a because-clause.

Because Turkish very often omits pronouns when the meaning is clear from context.

In this sentence, the natural understanding is that the person who was off is the same person as the main subject:

• Eniştem ... götürdü = My brother-in-law took ...
• so izinli olduğu için is understood as because he (my brother-in-law) was off

Turkish does this all the time. English often needs he/she/I/we, but Turkish often does not.

Götürdü is the definite past tense.

It means:

• he/she took
• he/she brought/took along depending on context, though here took is the natural translation

Breakdown:

• götür- = take, carry, bring along
• -dü = past tense

So:

• götürdü = he/she took

This tense is used for completed past events that the speaker presents as factual.

Because Turkish past tense endings change according to vowel harmony and voicing.

The past tense can appear as:

• -dı
• -di
• -du
• -dü
• or with t instead of d after certain voiceless sounds

In götürdü:

• the final vowel of the stem is ü, so the suffix vowel also becomes ü
• the stem ends in r, which is voiced, so the consonant is d, not t

So:

• götür + dü = götürdü

They are quite different:

• gitmek = to go
• götürmek = to take someone/something somewhere

So:

• Parka gitti. = He went to the park.
• Yeğenimi parka götürdü. = He took my niece/nephew to the park.

A useful pair to remember is:

• getirmek = to bring
• götürmek = to take

Yes. Turkish word order is flexible, although some orders sound more neutral than others.

The given sentence is natural:

• Eniştem bugün izinli olduğu için yeğenimi parka götürdü.

But you could also say:

• Bugün izinli olduğu için eniştem yeğenimi parka götürdü.

This also means the same thing: Because he was off today, my brother-in-law took my niece/nephew to the park.

Changing the order often changes focus or emphasis, not the basic meaning.

Yes. Bugün modifies the reason clause.

So the idea is:

• bugün izinli = off today
• bugün izinli olduğu için = because he was off today

It is not mainly describing götürdü (took today), but rather the reason: today he had the day off, so he took the child to the park.

Yes, but the structure changes.

With çünkü, you usually make two separate clauses:

• Eniştem yeğenimi parka götürdü, çünkü bugün izinliydi.

That also means My brother-in-law took my niece/nephew to the park because he was off today.

Difference:

• olduğu için = a built-in subordinate because structure
• çünkü = a separate conjunction, more like English because

Both are common, but X olduğu için is especially common in Turkish when giving reasons in a smooth, connected way.