A szervezeti probléma jelentős.

A is the definite article, like English the.

Hungarian has two forms of the definite article:

• a before a consonant sound
• az before a vowel sound

Since szervezeti begins with sz, which is a consonant sound, Hungarian uses a:

• a szervezeti probléma = the organizational problem

Szervezeti comes from szervezet, which means organization.

The ending -i often turns a noun into an adjective meaning related to, connected with, or belonging to that thing.

So:

• szervezet = organization
• szervezeti = organizational

In this sentence, szervezeti probléma means organizational problem.

Not exactly.

• a szervezeti probléma = the organizational problem
  meaning a problem connected with organization, structure, or management

• a szervezet problémája = the organization's problem
  meaning the problem that the organization has

So szervezeti is an adjective, while a szervezet problémája shows possession.

In Hungarian, the present-tense verb to be is usually omitted in the 3rd person when the predicate is a noun or an adjective.

So:

• A szervezeti probléma jelentős. = The organizational problem is significant.

There is no missing word here; this is the normal Hungarian pattern.

Compare:

• A probléma nagy. = The problem is big.
• A ház új. = The house is new.

Yes, but the structure would change.

In your sentence, jelentős comes after the noun phrase because it is the predicate adjective:

• A szervezeti probléma jelentős. = The organizational problem is significant.

If you put jelentős before the noun, it becomes part of the noun phrase:

• a jelentős szervezeti probléma = the significant organizational problem

So:

• after the noun phrase = is significant
• before the noun = significant as a describing adjective

Not in the way they do in many European languages.

Hungarian has no grammatical gender, so adjectives do not change for masculine, feminine, or neuter.

In a sentence like this, jelentős simply stays jelentős in the singular:

• A probléma jelentős. = The problem is significant.

With a plural subject, the predicate adjective can appear in the plural:

• A problémák jelentősek. = The problems are significant.

But there is still no gender agreement.

Yes, but it changes the emphasis.

The original sentence:

• A szervezeti probléma jelentős.

is a neutral statement, with a szervezeti probléma as the topic.

If you say:

• Jelentős a szervezeti probléma.

the meaning is close, but now jelentős gets more focus, as if you are stressing It is significant.

Hungarian word order is often about emphasis and information structure, not just grammar.

You would say:

• A probléma szervezeti.

That is different from your original sentence:

• A szervezeti probléma jelentős. = The organizational problem is significant.
• A probléma szervezeti. = The problem is organizational.

So in the original sentence, szervezeti describes probléma. In the second sentence, szervezeti is the predicate.

The accent marks are part of normal Hungarian spelling and affect pronunciation.

• á in probléma is a long vowel
• ő in jelentős is also a long vowel, and it is a different sound from o or ö

They do not show stress.

Hungarian stress normally falls on the first syllable of each word:

• A SZErvezeti PROBléma JElentős

A few useful spelling/pronunciation rules from this sentence:

• sz sounds like English s
• s sounds like English sh
• j sounds like English y