Képes vagyok dolgozni.

• képes = capable, able
• vagyok = I am
• dolgozni = to work

So the structure is literally Capable am I to work, which in natural English is I am able to work or I can work, depending on context.

Because vagyok is the 1st person singular form of the verb lenni (to be).

Here are the relevant forms:

• vagyok = I am
• vagy = you are (singular, informal)
• van = he/she/it is

So:

• Képes vagyok dolgozni = I am able to work
• Képes van dolgozni would be incorrect

Hungarian often leaves out subject pronouns because the verb ending already shows the subject.

In vagyok, the ending tells you the subject is I, so én is not necessary.

• Képes vagyok dolgozni = normal, natural
• Én képes vagyok dolgozni = possible, but én adds emphasis, something like I am able to work

Hungarian uses pronouns mainly for contrast, emphasis, or clarity.

Dolgozni is the infinitive of the verb dolgozik (to work).

The infinitive in Hungarian often ends in -ni, like:

• dolgozni = to work
• tanulni = to study
• menni = to go

In this sentence, dolgozni works like English to work after to be able:

• I am able to work
• Képes vagyok dolgozni

Hungarian has more than one way to talk about ability.

Képes vagyok dolgozni means I am capable of working or I am able to work. It often sounds a bit more explicit or slightly more formal than the most everyday way.

A very common alternative is:

• Tudok dolgozni = I can work

The difference is often this:

• tudok can mean practical ability or knowing how
• képes vagyok stresses being capable, in a condition to do it, or able under the circumstances

In many situations, both can translate as I can work, but they are not always identical in tone.

Yes, it is correct and natural, but it can sound a little more deliberate or formal than Tudok dolgozni.

It fits well in contexts like:

• physical ability: After resting, I am able to work
• mental/emotional ability: Today I am capable of working
• situational ability: Now I am able to work

So it is natural, but the exact feel depends on context.

Yes. Hungarian word order is flexible, and changing it can change the emphasis.

The neutral version is:

• Képes vagyok dolgozni

You might also see:

• Dolgozni képes vagyok = puts more focus on working
• Én képes vagyok dolgozni = emphasizes I
• Képes vagyok én dolgozni = possible in special contexts, often contrastive or emotional

For a learner, the basic version Képes vagyok dolgozni is the safest and most useful.

You normally put nem before the conjugated verb:

• Nem vagyok képes dolgozni. = I am not able to work.

This is the standard way to negate it.

Compare:

• Képes vagyok dolgozni. = I am able to work.
• Nem vagyok képes dolgozni. = I am not able to work.

You can use képes in different patterns, but with an action, the infinitive is the most straightforward choice:

• Képes vagyok dolgozni. = I am able to work.

Hungarian also uses:

• képes valamire = capable of something

For example:

• Képes vagyok erre. = I am capable of this.
• Képes vagyok a feladatra. is less natural than other ways of expressing it; Hungarian often prefers a different structure depending on the noun

For beginners, it is best to remember:

• képes vagyok + infinitive for able to do something

A rough pronunciation guide is:

• Képes ≈ KAY-pesh
• vagyok ≈ vaw-yok
• dolgozni ≈ DOL-goz-nee

A few useful notes:

• é is a long vowel, like ay in say, but held a bit longer
• s in Hungarian sounds like English sh
• gy is a soft consonant somewhat like the d in British duke or a softened dy
• stress in Hungarian usually falls on the first syllable of each word:
  • KÉ-pes
  • VA-gyok
  • DOL-goz-ni

Only the form of to be changes:

• Képes vagyok dolgozni. = I am able to work.
• Képes vagy dolgozni. = You are able to work. (singular informal)
• Képes dolgozni. = He/She is able to work.
• Képesek vagyunk dolgozni. = We are able to work.
• Képesek vagytok dolgozni. = You are able to work. (plural informal)
• Képesek dolgozni. = They are able to work.

Notice that in the plural, képes becomes képesek.