A születésnapi torta kész.

A is the definite article, so it means the.

Hungarian has two forms of the definite article:

• a before consonant sounds
• az before vowel sounds

So:

• a torta = the cake
• az alma = the apple

Here, torta begins with a consonant, so Hungarian uses a.

Születésnapi is an adjective meaning birthday in the sense of related to a birthday.

It is built from:

• születésnap = birthday
• -i = an adjective-forming ending meaning something like related to or of

So:

• születésnap = birthday
• születésnapi = birthday-related / for a birthday

That is why születésnapi torta means birthday cake.

Hungarian very often forms this kind of adjective with -i.

In Hungarian, the verb to be is often omitted in the present tense, third person singular when the predicate is a noun or adjective.

So:

• A torta kész. = The cake is ready.
• literally: The cake ready.

This is normal Hungarian grammar.

But the verb appears in other persons:

• Kész vagyok. = I am ready.
• Kész vagy. = You are ready.

And it also appears in other tenses:

• A torta kész volt. = The cake was ready.
• A torta kész lesz. = The cake will be ready.

Because kész is acting as the predicate adjective, not as a normal adjective inside the noun phrase.

Compare:

• a kész torta = the ready/finished cake
  Here, kész directly describes torta, so it comes before the noun.

• A születésnapi torta kész. = The birthday cake is ready.
  Here, kész is the main statement about the cake, so it comes after the noun.

So the position changes depending on the job the adjective is doing.

No. Torta is in the basic dictionary form here.

That is because it is the subject of the sentence, and Hungarian subjects in the singular usually have no special ending.

So:

• torta = cake
• a torta = the cake

There is no accusative ending here because this is not a direct object.

Yes, Hungarian word order is flexible, but changing it usually changes the emphasis.

The sentence you have:

• A születésnapi torta kész.

is a neutral, straightforward way to say it.

You could also say:

• Kész a születésnapi torta.

This puts more emphasis on ready—something like The birthday cake is ready now or Ready is the birthday cake in terms of focus.

So both are possible, but they do not feel exactly the same.

A rough English-friendly pronunciation would be:

aw SYU-le-taysh-na-pi TOR-taw kayss

A few important sounds:

• sz is pronounced like English s
• é is a long ay sound, but purer than in many English accents
• kész sounds roughly like kayss
• ü has no exact English equivalent; it is similar to German ü or French u

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

• A
• SZÜletésnapi
• TORta
• KÉSZ

There is a more casual spoken version:

• szülinapi

So you may also hear:

• A szülinapi torta kész.

This means the same thing, but szülinapi is more informal and conversational.

By contrast:

• születésnapi is the more standard and neutral form.

Not in standard Hungarian for this sentence.

Because Hungarian normally leaves out van in the present tense, third person singular with an adjective predicate, the natural form is:

• A születésnapi torta kész.

Using van here would sound wrong to a learner aiming for standard Hungarian.

There are some other expressions involving van and készen, but that is a different structure. For this sentence, the correct version is simply:

• A születésnapi torta kész.