A virágcserépben virágföld van, és vízelvezető nyílása is van.

A virágcserépben means in the flowerpot.

It breaks down like this:

• virág = flower
• cserép = pot, flowerpot
• virágcserép = flowerpot
• -ben = in

So virágcserépben literally means in flowerpot, and with a it becomes in the flowerpot.

This is because of vowel harmony.

Hungarian has two versions of many endings:

• -ban after back-vowel words
• -ben after front-vowel words

Here the last part of the compound is cserép, which has front vowels, so Hungarian uses -ben:

• virágcserépben = in the flowerpot

This is a very common pattern in Hungarian.

Because virágföld is being used as a bare noun, which is very natural in Hungarian, especially in an existential sentence like this.

Here virágföld means potting soil / flower soil, which is a material or substance, not a countable item. Hungarian often leaves out an article in this kind of case:

• A virágcserépben virágföld van. = There is potting soil in the flowerpot.

Using egy would sound odd here, because virágföld is not really one potting soil in English either.

Van is the 3rd person singular form of lenni = to be.

In this sentence it works like there is / there exists:

• A virágcserépben virágföld van.
  = There is potting soil in the flowerpot.

Hungarian very often uses van in sentences that English translates with there is.

It also uses van in structures that English translates with have, which is what happens in the second clause.

Hungarian word order is strongly influenced by topic and new information.

Here:

• A virágcserépben = the topic, what we are talking about
• virágföld van = the new information about it

So the sentence is roughly:

• As for the flowerpot, there is potting soil in it.

This is very natural Hungarian word order. Another order can also be possible, but it would shift the emphasis.

Vízelvezető means something like drainage or water-draining.

It is built from:

• víz = water
• elvezet = lead away, drain away
• elvezető = draining / conducting away

So vízelvezető nyílás is a drainage opening or drain hole.

Hungarian often creates long compound words like this, and they are very common in everyday language.

Because -a here is a possessive suffix.

• nyílás = opening
• nyílása = its opening

So vízelvezető nyílása means:

• its drainage opening
• or more naturally in English, a drainage hole/opening

The possessor is not named directly in that clause, but from the context we understand that it is the flowerpot.

Because Hungarian does not normally use a separate verb equivalent to English have in this kind of sentence.

Instead, Hungarian says something like:

• there is its X

So:

• vízelvezető nyílása van literally means
• it has a drainage opening but more literally
• its drainage opening exists

This is a very important Hungarian pattern:

• Van autóm. = I have a car.
• Nyílása van. = It has an opening.

Here is means also or too.

So:

• vízelvezető nyílása is van means
• it also has a drainage opening

In Hungarian, is usually comes after the word or phrase it belongs to.

So:

• nyílása is = its opening too / also its opening

English usually puts also in a different place, but Hungarian places is directly after the relevant element.

Because the sentence has two separate clauses, and each one has its own predicate:

1. A virágcserépben virágföld van
2. és vízelvezető nyílása is van

So Hungarian repeats van naturally:

• there is potting soil
• and it also has a drainage opening

Leaving out the second van would make the sentence incomplete.

Yes, absolutely.

That version makes the possessor explicit:

• A virágcserépnek vízelvezető nyílása is van.
• The flowerpot also has a drainage opening.

Here -nak/-nek marks the possessor:

• a virágcserépnek = to the flowerpot / of the flowerpot, but in this structure it functions like the flowerpot has

The original sentence leaves this possessor unstated because it is already clear from the context.