Herkes yeni kralı alkışladı.

Because kralı is the direct object of the sentence, and in Turkish a specific/definite direct object usually takes the accusative ending.

• kral = king
• kralı = the king / the specific king as the object

So:

• Herkes yeni kralı alkışladı. = everyone applauded the new king

If you removed the accusative ending, yeni kral would sound less definite and would not fit this sentence as naturally.

Because Turkish adjectives do not agree with the noun in number, gender, or case.

So:

• yeni kral = new king
• yeni kralı = the new king (object form)

Only the noun changes here, not the adjective:

• yeni stays yeni
• kral becomes kralı

This is much simpler than in many European languages.

Because herkes means everyone/everybody, and in Turkish it is treated grammatically as singular.

So the verb is in third person singular:

• Herkes ... alkışladı. = Everyone ... applauded.

This is similar to English in a way:

• Everyone applauded. not
• Everyone applaudeded / applaud etc.

So even though the meaning includes many people, the grammar is singular.

Alkışladı can be broken down like this:

• alkışla- = to applaud / clap for
• -dı = past tense
• third person singular has no extra ending after that

So:

• alkışladı = he/she/it applauded
• in this sentence: everyone applauded

You may also see the dictionary form:

• alkışlamak = to applaud

Because the most neutral Turkish word order is usually:

Subject + Object + Verb

So here:

• Herkes = subject
• yeni kralı = object
• alkışladı = verb

That gives the standard order:

• Herkes yeni kralı alkışladı.

Turkish word order can change for emphasis, but this version is the most straightforward and neutral.

For example, other orders are possible in context, but they would sound more marked:

• Yeni kralı herkes alkışladı.
• Alkışladı herkes yeni kralı. (much more stylistically marked)

Not here.

This is a very common learner question because kralı can look like a possessive form in some contexts. But in this sentence, kralı is best understood as:

• kral + ı
• king + accusative
• so: the king as a definite object

Why not his king?

Because his/her king would normally need context, and the sentence structure here clearly points to the accusative object reading.

So here:

• yeni kralı = the new king not
• his new king

Turkish does not have a separate word exactly like English the.

Instead, definiteness is often shown in other ways, especially with the accusative ending on direct objects.

So in this sentence:

• kralı helps show that this is a specific king
• that is why English translates it naturally as the new king

Turkish often leaves things that English marks with articles to be understood from:

• case endings
• context
• word order

There are two main clues:

1. Herkes is a subject-type word meaning everyone
2. kralı has the accusative ending, which strongly signals a direct object

So the structure is:

• Herkes = who did the action
• yeni kralı = who received the action
• alkışladı = the action

That makes the sentence easy to parse as:

Everyone applauded the new king.

It can mean both, depending on context, but in a sentence like this it is best translated as to applaud.

• alkışlamak = to clap for someone / to applaud someone
• alkış = applause / clapping

So:

• Herkes yeni kralı alkışladı. is more naturally
• Everyone applauded the new king than
• Everyone clapped the new king

In English, applaud is the better match when the object is a person.

Because of vowel harmony.

The accusative ending has four forms:

• -ı
• -i
• -u
• -ü

Which one you use depends on the last vowel of the noun.

In kral, the last vowel is a, so the correct accusative form is:

• kral + ı = kralı

This follows Turkish vowel harmony patterns.

A rough pronunciation guide:

• Herkes → HER-kes
• yeni → YE-ni
• kralı → kra-LUH (the final ı is the Turkish unrounded vowel, not exactly English ee)
• alkışladı → al-kush-LA-duh approximately

A few helpful notes:

• Turkish ı is not the same as English i
• ş sounds like sh
• Each letter is pronounced quite consistently in Turkish

So the whole sentence is roughly:

HER-kes YE-ni kra-LUH al-kush-LA-duh

That is only approximate, but it helps get started.