A figyelem nagyon fontos.

In Hungarian, a is the definite article, so here it means the, not English a/an.

• a is used before most consonant sounds
• az is used before vowel sounds

So:

• a figyelem = the attention
• az ember = the person / the human

In a sentence like this, Hungarian often uses the definite article with abstract nouns more naturally than English does, so A figyelem nagyon fontos is perfectly normal.

Figyelem most commonly means attention.

Depending on context, it can refer to:

• paying attention
• attentiveness
• notice or awareness

You may also see it in expressions such as:

• Figyelem! = Attention! / Warning!
• figyelni = to pay attention / to watch / to observe

So in this sentence, figyelem is a noun meaning attention.

Hungarian often leaves out the verb to be in the present tense, third person singular and plural.

So:

• A figyelem nagyon fontos. = Attention is very important.
• literally: The attention very important.

This is normal Hungarian grammar.

Compare:

• Én fáradt vagyok. = I am tired.
• Te fáradt vagy. = You are tired.
• Ő fáradt. = He/She is tired.

Notice that in the third person (ő), the present-tense is disappears.

Yes, Hungarian word order is flexible, but the neutral, straightforward order here is:

• A figyelem = topic / what we are talking about
• nagyon fontos = what we say about it

So this sentence has a natural topic + comment structure:

• As for attention, it is very important.

You can change word order for emphasis, but that changes the focus somewhat. For example:

• Nagyon fontos a figyelem.
  This sounds more like Attention is very important with extra emphasis on very important.

The version you were given is a simple, standard way to say it.

Nagyon means very.

It intensifies adjectives and adverbs:

• fontos = important
• nagyon fontos = very important

Other examples:

• nagyon jó = very good
• nagyon gyors = very fast
• nagyon szépen = very nicely

So here nagyon simply strengthens fontos.

No. Hungarian adjectives do not agree with nouns in gender, because Hungarian has no grammatical gender.

So fontos stays the same whether the noun is:

• masculine in English meaning
• feminine in English meaning
• singular
• plural

Examples:

• A könyv fontos. = The book is important.
• A kérdés fontos. = The question is important.
• A kérdések fontosak. = The questions are important.

Notice one important detail: when the adjective stands as the predicate, it can take a plural ending:

• singular: fontos
• plural: fontosak

But there is still no gender change.

A rough pronunciation is:

FIG-yeh-lem

A few helpful notes:

• Hungarian stress is always on the first syllable: FI
• gy is a single Hungarian consonant sound, somewhat like the dy in British duke or a softened g
• e is usually pronounced like eh

So figyelem sounds roughly like:

• FI-dyeh-lem

Not perfect, but close enough for a learner.

A rough guide is:

Ah FIG-yeh-lem NAH-dyon FON-tosh

A few pronunciation tips:

• a = short aw/ah
• gy in figyelem and nagyon is a soft consonant
• j in Hungarian sounds like English y
• s in Hungarian sounds like English sh

So:

• fontos is pronounced roughly FON-tosh, not fon-tos

And remember: the stress in each word is on the first syllable:

• A
• FIgyelem
• NAgyon
• FONtos

Both are possible in Hungarian, but they do not always feel exactly the same.

• A figyelem nagyon fontos. = Attention is very important.
• Figyelem! by itself often means Attention! as a warning or announcement.

Using a makes figyelem clearly function as the noun phrase the attention in a general statement. It sounds complete and natural in this kind of sentence.

So for a general statement about attention as a concept, A figyelem nagyon fontos is the safest and most natural version.

It is neutral and completely standard.

It could be used in:

• everyday conversation
• teaching
• advice
• writing
• presentations

It is not especially formal or especially casual. It is just a normal Hungarian sentence.

Yes.

• A figyelem fontos. = Attention is important.
• A figyelem nagyon fontos. = Attention is very important.

The only difference is the strength of the statement.
Adding nagyon makes it stronger.