A körülmény fontos.

A is the Hungarian definite article, so here it means the.

Hungarian has two forms of the definite article:

• a before a consonant
• az before a vowel

Since körülmény begins with k, which is a consonant, the correct form is a körülmény.

Examples:

• a ház = the house
• az autó = the car

In Hungarian, the verb to be is usually omitted in the present tense, third person when the sentence says that something is something or is some quality.

So:

• A körülmény fontos. = The circumstance is important.

But Hungarian does include van or another form of to be in many other situations, for example:

• Én fáradt vagyok. = I am tired.
• Te fáradt vagy. = You are tired.
• A körülmény fontos volt. = The circumstance was important.

So the missing is is completely normal here.

Great question. In Hungarian, adjectives come before the noun when they directly describe it inside a noun phrase:

• a fontos körülmény = the important circumstance

But in your sentence, fontos is not simply modifying the noun inside the phrase. It is the predicate adjective: it tells you something about A körülmény.

So the structure is:

• A körülmény = the subject
• fontos = what is being said about the subject

That is why the sentence means:

• The circumstance is important.

So compare:

• a fontos körülmény = the important circumstance
• A körülmény fontos. = The circumstance is important.

A rough pronunciation is:

• körülmény ≈ kuh-rueel-maayn
  But that English-style spelling is only approximate.

A few important points:

• Hungarian stress is almost always on the first syllable: KÖrülmény
• ö is a rounded vowel, somewhat like the vowel in German schön or French peu
• ü is also a rounded front vowel, like German ü in über
• ny is pronounced like the ny in Spanish señor or the ni in onion

So the ending -mény sounds roughly like myaayn, with ny as a single sound.

Here körülmény is in the nominative case, which is the basic dictionary form. It is the subject of the sentence.

Hungarian often uses no ending at all for the nominative singular, so:

• körülmény = circumstance / the circumstance depending on context and article

Because it is the subject, no special case ending is needed here.

Yes. Fontos a körülmény is also grammatical.

Both sentences mean essentially the same thing, but the word order changes the emphasis.

• A körülmény fontos. puts the topic first: as for the circumstance, it is important
• Fontos a körülmény. emphasizes important

Hungarian word order is often influenced by topic and focus, not just by fixed subject-verb-object rules.

So:

• A körülmény fontos. = neutral, topic-first
• Fontos a körülmény. = stronger emphasis on important

Körülmény is singular.

To make it plural, Hungarian adds -ek here:

• körülmények = circumstances

Then the predicate adjective also becomes plural:

• A körülmények fontosak. = The circumstances are important.

So compare:

• A körülmény fontos. = The circumstance is important.
• A körülmények fontosak. = The circumstances are important.

Sometimes yes, sometimes no.

When fontos comes before a noun as a normal adjective, it does not change for singular or plural:

• a fontos körülmény = the important circumstance
• a fontos körülmények = the important circumstances

But when fontos is used as a predicate adjective, it does take a plural ending if the subject is plural:

• A körülmény fontos.
• A körülmények fontosak.

So in predicate position, plural subjects usually require plural adjective forms.

Yes, it is grammatical and natural, especially in a context where one particular circumstance is being discussed.

For example, if several factors are being considered, someone might say:

• A körülmény fontos.

meaning that this specific circumstance matters.

Without context, it may sound a little formal or abstract, because körülmény is a somewhat formal word meaning circumstance or factor. But the sentence itself is perfectly normal.