In English, the words this and that (when not used as a conjunction) can be thought of as having two meanings each.

1. I like this cat. That dog bit me. This girl is beautiful. That statement is false.
2. I like this. You liked that. This is beautiful. That is not true.

1. A more specific version of THE. "I like the cat" can be made more specific by saying "I like this/that cat".
2. A standalone pronoun that is NOT a substitute for THE. We could not say "I like the", but we can say "I like this/that."

It's annoying that in English the word "this" is identical in these two cases.

In Hungarian, of course, it's more logical. The two uses of "this" are translated differently:

1. ez a; az a this; that
2. ez; az this; that

Before we apply any endings, let's look at the two versions in nominative (dictionary) form.

1. Ez a macska fehér This cat is white.
   Az a macska fekete That cat is black.
   Ez az alma piros This apple is red. N.B the vowel means a becomes az (like a becomes an in English)
   Az az alma zöld That apple is green.
2. Ez macska This is a cat.
   Az kutya That is a dog.
   Remember how we drop van from these, so in reality we're saying ez macska (van) ; az kutya (van).

Now let's put some endings onto the (pro) nouns.

The easiest is the accusative. Remember, we apply -t to the object of a verb.

• Én szeretem Ágit I like Ági.
• Én ettem almákat I ate apples.
• Én látom a kutyát I see the dog.

1: When the object is not a single word (Ági, alma, kutya) but has this/that prefixing it, i.e. is case (1) above, when we're using it as a more specific form of the, then we must apply the ending to both parts:

• Én látom ezt a kutyát I see this dog.
• Én látom azt a kutyát I see that cat.

2: When we use this/that as a standalone pronoun, we apply endings to the (only) word

• Én szeretem ezt I like this/it.
• Én szeretem azt I like that.

Assimilation of endings is very important. As we have seen, we apply an ending to the noun itself and also to the qualifiying pronoun:
Én látom ezt a kutyát I see this dog.

It's easy to see that we simply add the -t to both the ez/az and the noun.

However, the above is the simplest case. Most other noun cases require that the ez/az is modifed in some way.

Let's take -ban as an example. We want to say, "I live in this house." That is our objective.

• ház house
• ez a ház this house
• én lakom I live (reside)
• In = -ban/ben

Now we must start with ez a ház this house and apply -ban/ben to it, to give us in this house.

We do this by putting -ban or -ben on both the ez and on the noun:

• ez+(ban/ben)      a      ház+(ban/ben)

(Of course we choose either ban OR ben based on vowel harmony).

Let's adjust the ház first: The correct choice is -ban, because ház is a back-vowel word. So far we have:

• ez+(ban/ben)      a      házban

Now we must adjust ez. The correct choice of -ban or -ben is -ben, because this agrees with ez. N.B. our choice of harmonious variant for the ez/az is independent of our choice for the noun itself!

We have so far:

• ez+ben a házban

This is when the assimilation occurs. The -z in ez/az changes into the first consonant of the ending!

• ez becomes eb (from ben)

We then put in the -ben to get ebben.

Overal, the change is:

• ez + ben = ebben
• ebben a házban

Altogether: Én lakom ebben a házban. QED.

If we wanted to say, "I live in that house," we would start with

• az a ház
• az+ban = abban
• Én lakom abban a házban.

Here is a list the main noun endings, showing how the final -z in ez/az assimilates into the first consoonant of the new ending.

• eddig/addig until now/then
• ekkor/akkor at this/that time
• ebbe/abba into this/that
• ebből/abból out of this/that
• ezért/azért for this/that reason (herefore/therefore)
• ehhez/ahhoz towards this/that
• ennek/annak for/to this/that
• ennél/annál by/at this/that
• erre/arra onto this/that
• erről/arról from/about this/that
• ettől/attól from this/that
• evvé/avvá (or ezzé/azzá) turning into this/that
• evvel/avval (or ezzel/azzal) with this/that

We also have ezen/azon on this/that but there is no assimilation here becase the ending, -on/en, doesn't start with a consonant (a requirement for the assimilation).

• Lakom abban a városban I live in that city
  
• Laksz ebben a városban You live in this city
  
• Lakom abban a épületben I live in that building
  
• Laksz ebben a épületben You live in this building
  

Note how when the -ban/-ben is applied to ez/az, its vowel harmonises with ez/az, not with the final object.

úsztam a tengerben. Benne, láttam halakat. I swam in the sea. In it, I saw fish.

Of course, ez and az can receive endings without needing a noun (like in situation (2) above). Then, the same rule applies to ez/az.

• Én lakom ebben I live in this
• Én lakom abban I live in that
• Ági beszél erről Ági speaks about this
• Ági beszél arról  Ági speaks about that

We apply the plural to ez/az as normal:

1. More specific case of the (plural):

• ezek a ... -k these
• azok a ... -k those

• ezek a kutyák fehérek these dogs are white
• azok a kutyák feketék those dogs are black

Of course they can still take noun case endings exactly like the singular brothers:

• Én látom ezeket a kutyákat I see these dogs
• Én látom azokat a házakat I see those houses
• Mi lakunk ezekben a házakban We live in those houses
• Ő laknak azokban a házakban They live in those houses

N.B. The assimilation does not happen for ezek/azok
The assimilation that we saw happening to the -z in ez/az does NOT happen to the trailing -k in ezek/azok. They simply take the noun endings without any change (execpt for endings starting with -v- as normal)

• ezekhez/azokhoz toward these/those
• ezeknél/azoknál by/at these/those
• ezekben/azokban in these/those
• ezekkel/azokkal with these/those
  N.B. the effect visible here is not the assimilation of the -k in ezek/azok, rather it's the gemination of the -v- in -val/vel .

2. Standalone noun

• ezek these
• azok those

Standalone nouns can take endings too:

• Szereted ezeket You like these
• Szereti azokat He likes those

The word "that" can be used as a conjuction: I see that you are here. It doesn't matter that you're late. This is given by the ever-present Hungarian word, hogy. Click to see the page about hogy.