Az ajándékcsomag az asztalon van.

Because there are two separate noun phrases, and each one is definite:

• az ajándékcsomag = the gift package
• az asztalon = on the table

Hungarian uses the definite article with both of them, just as English can say the gift package is on the table.

Also, Hungarian has two forms of the definite article:

• a before a consonant sound
• az before a vowel sound

So we get:

• az ajándékcsomag because ajándékcsomag starts with a vowel sound
• az asztalon because asztalon starts with a vowel sound

Asztalon is asztal + -on.

• asztal = table
• -on = a case ending meaning on in the sense of on the surface of

So Hungarian often uses a suffix where English uses a separate preposition.

Here:

• asztalon = on the table

This case is called the superessive case. Its forms are usually:

• -on
• -en
• -ön

Which one appears depends partly on the word and vowel harmony.

Hungarian word order is more flexible than English word order. It is not based only on subject-verb-object order. It also reflects topic, focus, and what information is already known or being emphasized.

In Az ajándékcsomag az asztalon van:

• Az ajándékcsomag is the topic: what we are talking about
• az asztalon van tells us where it is

So the sentence feels like:

• As for the gift package, it is on the table

This is a very natural neutral sentence in Hungarian.

In this sentence, van is the verb to be, and it comes at the end because that is a normal position in a neutral Hungarian sentence like this one.

The structure is roughly:

• Topic: Az ajándékcsomag
• Location phrase: az asztalon
• Verb: van

So literally the order is something like:

• The gift package on the table is

That may feel unusual to an English speaker, but it is normal in Hungarian.

No, not in this sentence.

Hungarian often drops the verb to be in the present tense, third person when the predicate is a noun or adjective:

• Az ajándékcsomag szép. = The gift package is nice.
• Ez ajándék. = This is a gift.

But with a location, Hungarian normally keeps van:

• Az ajándékcsomag az asztalon van.

So you cannot normally say just Az ajándékcsomag az asztalon with the same standard meaning.

It is a compound noun.

It is made from:

• ajándék = gift
• csomag = package / parcel

Together, ajándékcsomag means something like:

• gift package
• gift parcel
• sometimes gift pack

Hungarian forms compounds very freely, much more often than English does.

Because the sentence refers to a specific table: the table.

Hungarian, like English, can use a definite article when something is definite or identifiable.

So:

• asztalon = on a table / on table depending on context, but usually article-less forms are used differently and sound incomplete in many basic sentences
• az asztalon = on the table

In this sentence, the normal form is az asztalon.

Yes. That is also grammatical, but the emphasis changes.

• Az ajándékcsomag az asztalon van = neutral, talking about the gift package and saying where it is
• Az asztalon van az ajándékcsomag = puts more attention on the table / the location

This second version can feel more like:

• The gift package is on the table
• or, in a more contrastive context, It is on the table that the gift package is

So both are possible, but they are used in slightly different contexts.

Not exactly. This sentence more directly means:

• The gift package is on the table

Because az ajándékcsomag is definite: the gift package.

If you want to say There is a gift package on the table, Hungarian would more naturally use an indefinite noun and a different structure, for example:

• Van egy ajándékcsomag az asztalon.

That means:

• There is a gift package on the table.

So the sentence you were given is about the location of a known object, not just its existence.

A few useful pronunciation points:

• sz is pronounced like English s
• cs is pronounced like English ch
• gy is a soft sound, somewhat like the dy in duke in some pronunciations, but softer
• j is like English y
• a in Hungarian is more like o in British hot
• á is a longer, brighter vowel

So roughly:

• Az sounds like awz
• ajándékcsomag sounds roughly like aw-yaan-dayk-cho-mag
• asztalon sounds roughly like aws-taw-lon
• van sounds roughly like von

These are only approximations, but they help at the beginning.

Asztalon is in the superessive case, which expresses being on top of something, usually on a surface.

You do not absolutely need the case name at first, but it can be useful because Hungarian grammar is often taught through case endings.

Here is the basic idea:

• asztal = table
• asztalon = on the table
• asztalra = onto the table
• asztalról = off the table

So even if you do not memorize every case name immediately, it is very helpful to notice the pattern:

• place on something
• movement onto something
• movement off something

Yes. It is a perfectly normal Hungarian sentence.

It is simple, neutral, and natural. A Hungarian speaker would easily say this in everyday speech when answering a question like:

• Hol van az ajándékcsomag? = Where is the gift package?

Then the answer:

• Az ajándékcsomag az asztalon van.

would sound completely normal.