In English you always need to use any pronoun (direct, indirect, demonstrative…) irregardless of whether the verb refers to an indefinite or definite object/person.

I’m writing a letter. I’m writing the letter.

The object is ’letter’ and our verb ’write’ is the same in both sentences. The indefiniteness and definiteness of ’letter’ are expressed with the indefinite and definite articles ’a, the’. Here’s the Hungarian translation:

I’m writing a letter. – Írok egy levelet.
I’m writing the letter. – Írom a levelet.

As you can see, the Hungarian verb ’ír-write’ has two different conjugations. The indefinite conjugation requires the -ok suffix in 1st person singular present tense, whereas the definite conjugation in the same person/number/tense is formed with the -om suffix. The indefinite and definite articles are used accordingly.

So far so good. But what if we get a question like ’Are you writing a/the letter’? How do you answer that question?

Are you writing a letter? – Yes, I’m writing it.
Írsz egy levelet? – Igen, azt írok.

Are you writing the letter? – Yes, I’m writing it.
Írod a levelet? – Igen, azt írom.

The pronoun ’it’ refers back to ’a letter’ and ’the letter’. In the same way, the Hungarian demonstrative pronoun ’az’ turns into an accusative pronoun ’azt’ to refer back to ’egy levelet’ and ’a levelet’.
And that’s when omitting pronouns becomes relevant. In everyday language, we do tend to ”forget” saying certain pronouns when the context is clear and we know for sure what we are talking about. So the above-mentioned answers can be turned into:

Igen, írok. Igen, írom.

It is more common, though, that we leave out pronouns when the verbs are in definite conjugation. Since the definite conjugation already refers to the object, there is no confusion about the context. Let’s see more examples with other pronouns, too. So that you know what I omit, I’ll parenthesize the pronouns. It is also evident from the examples that English has to use those pronouns.

-Látod a lányt? – Igen, látom (őt).
-Can you see the girl? – Yes, I can see her.

-Akarjátok az új ruhákat vagy sem? – Nem, nem akarjuk (azokat).
-Do you want the new clothes or not? – No, we don’t want them.

The problem starts when there seems to be no context like in this question:

-Látod? – Can you see it?

We translate it with ’can you see it’ because ’látod’ is obviously in definite conjugation. The speaker knows exactly what he sees and that’s why he’s asking ’Látod?’.

-Látod? – Can you see it?
-Semmit nem látok. Te mit látsz? – I can’t see anything. What can you see?
-A boltot. – The shop.
-Igen, már látom. – Yes, I can see it now.

The other person answers ’nem látok’ in indefinite conjugation because he cannot see anything. And ’anything/nothing’ is something indefinite. So is ’something’, by the way :). Then he asks ’mit látsz’ in indefinite conjugation because he still cannot see anything. Finally, the speaker clarifies ’boltot’. So the other person answers ’látom’ in definite conjugation. Now he knows exactly what he sees. It does not matter if the answer is affirmative or negative. The same rules apply.

That’s why you give such answers:

-Nem értek semmit. -Az egyenletet így kell megoldani. Már érted? -Igen, értem.
-I don’t understand anything. -The equation has to be solved like this. Do you understand now? -Yes, I understand (it).

This is a good example for native English speakers because as you see the verb ’understand’ does not require ’it’ when you answer. Yet you know exactly what you understand.

-Érted? – Do you understand?
-Értem. – I understand.

Let’s take a look at more examples:

-A bank elveszi a házadat. – Nem hagyom.
-The bank is going to take your house. – I’m not going to let (it happen).

-Tessék a visszajáró! – Köszönöm.
-Here’s your change. – Thank you.

The verb ’köszön’ has two meanings ’to say thank you’ and ’to greet’. If it means ’to say thank you’, it is transitive, so Hungarian people thank something, and not thank for something. On the other hand, if you greet someone, it is transitive in English, but it requires an indirect object in Hungarian. Actually, we say ’greet to someone = köszön valakinek’.

-Köszönöm a visszajárót! – Köszöntem önnek, amikor bejöttem? – Igen, ön mindig köszön nekem.
-Thank you for the change. – Did I greet you when I entered? – Yes, you always greet me.

The difficulty also lies in the different use of verbs in English and Hungarian. A verb that is transitive in English might be intransitive in Hungarian and viceversa. In the above-mentioned sentence you can’t use ’köszön’ in definite conjugation because it is not transitive. It cannot require an object.
Let’s contrast indefinite and definite conjugation with ’ért’.

-Érted? – Mindent értek.
-Do you understand? – I understand everything.

’Minden’ is indefinite or general, so ’értek’ is in indefinite conjugation.

PROBLEMS WITH ACCUSATIVE PRONOUNS

As languages do not consist of rules that always make sense, we have to be prepared to learn odd things. That is the case with accusative pronouns if you conjugate verbs in definite mode. To make this problem tangible, I’ll use the accusative pronouns ’őt’ and ’engem’.

Látom őt. – I can see him.
Látod őt. – You can see him.
Látja őt. – He can see him.
Látjuk őt. – We can see him.
Látjátok őt. – You can see him.
Látják őt. – They can see him.

’Lát’ is in definite conjugation in every number/person. But what if we use ’engem’?

Lát engem. – He can see me.
Lát téged. – He can see you.
Látja őt. – He can see him.
Lát minket. – He can see us.
Lát titeket. – He can see you.
Látja őket. – He can see them.

’Lát’ is used in definite conjugation only when referring to the accusative pronoun in 3rd person singular and 3rd person plural. In any other number/person (engem, téged, minket, titeket) ’lát’ is in indefinite conjugation.

This goes for ’őt, őket, önt, önöket, maga, magát’. That is, for polite forms, too.

If there is an accusative pronoun, one would think it is definite. Well, it’s no use asking why it is that way. It just is. I couldn’t find any explanation why this phenomenon had developed the way it is. Something for you to struggle with :). So let’s ask more questions and give the right answers.

-Érted a szabályt? – Igen, értem (azt).
-Do you understand the rule? – Yes, I understand (it).

-Érted őket? – Igen, értem (őket).
-Do you understand them? – Yes, I understand them.

-Értesz engem? – Igen, értelek (téged).
-Do you understand me? -Yes, I understand you.

-Látjátok őket? – Nem, nem látjuk (őket).
-Can you guys see them? – No, we can’t see them.

-Látnak minket? – Nem, nem látnak (minket). Önt viszont látják.
-Can they see us? – No, they can’t see us. However, they can see you, sir.