A modem és a vezeték az asztalon van.

They are the same word: the Hungarian definite article, meaning the.

• Use a before a consonant sound: a modem, a vezeték
• Use az before a vowel sound: az asztalon

So this is just a pronunciation-based alternation, like choosing the easier form to say.

Hungarian usually uses a case ending instead of a separate preposition.

• asztal = table
• -on = a suffix meaning on

So:

• asztalon = on the table

With the article included, az asztalon literally looks like the table-on.

Because in Hungarian, the article and the case ending do different jobs:

• az = the
• asztalon = on the table

The case ending goes on the noun, not on the article. So Hungarian says:

• az asztalon = on the table

This is completely normal.

That is true only in certain kinds of sentences.

In present-tense third person, Hungarian often drops van when the predicate is a noun or adjective:

• A modem új. = The modem is new.

But with location, van is normally kept:

• A modem az asztalon van. = The modem is on the table.

So in this sentence, van is expected because the sentence tells us where something is.

This is a very good question.

In careful standard Hungarian, many speakers would prefer:

• A modem és a vezeték az asztalon vannak.

because the subject is effectively plural: the modem and the wire.

However, singular van can sometimes be heard in everyday speech, especially when the two things are felt as a kind of set or grouped unit.

So for a learner:

• safest standard choice: vannak
• the version with van: possible in colloquial usage, but not the best default to imitate in formal Hungarian

Hungarian word order is more flexible than English word order. It often shows topic and emphasis, not just grammar.

In:

• A modem és a vezeték az asztalon van

the sentence starts with what we are talking about: the modem and the wire, and then says where they are: on the table.

You could also say:

• Az asztalon van a modem és a vezeték.

That version puts more attention on the location, as if answering What is on the table? or emphasizing on the table.

So the words can move, but the emphasis changes.

In normal Hungarian, if both nouns are definite, each one usually gets its own article:

• a modem és a vezeték

That is the most natural standard form.

If you leave out the second article, it can sound less natural or more like a compressed label. So learners should normally repeat the article with each definite noun.

No. Here modem is just the noun itself. The m is part of the word.

So:

• a modem = the modem

If you wanted to say my modem, you would say:

• a modemem
• or more explicitly az én modemem

So this m is not a possessive ending here.

Not always exactly. Vezeték is a broader word that can mean things like:

• wire
• cable
• line
• sometimes lead

So the best English translation depends on context. In a simple sentence like this, wire is fine, but in other situations cable might sound more natural.