A fogfájás súlyos.

In Hungarian, a and az are the definite article, so they mean the.

• a is used before words that begin with a consonant: a fogfájás
• az is used before words that begin with a vowel: az alma

So A fogfájás súlyos. means The toothache is severe., not A toothache is severe.

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

So:

• A fogfájás súlyos. = The toothache is severe.

There is no separate word for is here.

But in other persons or tenses, the verb appears:

• Súlyos vagyok. = I am serious / severe (grammatically, though not a natural example in meaning)
• A fogfájás súlyos volt. = The toothache was severe.

This missing is is one of the most important basic patterns in Hungarian.

Fogfájás is a compound word:

• fog = tooth
• fájás = pain, aching

So literally it is something like tooth-pain or toothache.

This is very common in Hungarian: words are often built by combining smaller words into one compound.

Good question. Here, súlyos is not just describing the noun inside a noun phrase; it is the predicate of the sentence.

So:

• A fogfájás súlyos. = The toothache is severe.

But if you put the adjective before the noun, you get a noun phrase instead:

• a súlyos fogfájás = the severe toothache

So the difference is:

• A fogfájás súlyos. = a full sentence
• a súlyos fogfájás = just a noun phrase

A rough pronunciation is:

aw fok-faayaash soo-yosh

A few useful notes:

• fogfájás is roughly fok-faa-yaash
• súlyos is roughly soo-yosh
• the á sound is long
• the ú sound is long
• Hungarian stress is usually on the first syllable of each word

So the stress pattern is roughly:

• A FOGfájás
• SÚLyos

Also, in normal pronunciation, the g in fogfájás often sounds more like k because of the following f.

In this sentence, súlyos means severe or serious.

It can be used in several related senses:

• súlyos betegség = a serious illness
• súlyos hiba = a serious mistake
• A fogfájás súlyos. = The toothache is severe.

So the core idea is something serious, grave, severe, heavy in impact.

In a medical context, severe is usually the best translation.

Yes. That is also grammatical.

Hungarian word order is flexible, and changing the order often changes the focus or emphasis.

• A fogfájás súlyos. = neutral statement: The toothache is severe.
• Súlyos a fogfájás. = puts more emphasis on severe

So both are possible, but the first is the more neutral basic version for a learner.

Because both words are in their basic dictionary form here.

• fogfájás is the subject, in the nominative singular
• súlyos is a predicate adjective

Hungarian uses lots of endings, but not every sentence needs them. This sentence is structurally very simple:

• A fogfájás = the toothache
• súlyos = severe

No extra case ending is needed.

Yes, fogfájás is a completely normal word.

But in everyday speech, a Hungarian person might also say things like:

• Fáj a fogam. = My tooth hurts.
• Nagyon fáj a fogam. = My tooth hurts a lot.
• Erős a fogfájás. = The toothache is strong / severe.

So A fogfájás súlyos. is correct, but in real conversation people may often choose a more natural symptom-style expression such as Fáj a fogam.

They are related, but they are not the same structure.

• súlyos fogfájás = severe toothache
  This is just a noun phrase.

• A fogfájás súlyos. = The toothache is severe.
  This is a complete sentence.

So:

• adjective before noun = description inside a noun phrase
• adjective after noun = often part of the sentence predicate

This is a very useful distinction to learn early in Hungarian.