L'argilla è sul tornio da vasaio.

Because la becomes l' before a feminine singular noun that starts with a vowel.

• la argilla → l'argilla
• This is just elision, done for smoother pronunciation.

So l'argilla means the clay.

Italian uses articles more often than English.

In this sentence, l'argilla refers to a specific clay that is being talked about, so the clay is natural in Italian.

More generally, Italian often keeps the article with nouns where English might not:

• L'acqua è fredda. = The water is cold.
• Il pane è buono. = Bread is good.

So even if English sometimes drops the article, Italian often does not.

The accent distinguishes è (is) from e (and).

• è = is
• e = and

So here:

• L'argilla è sul tornio... = The clay is on the wheel...

Without the accent, it would be a different word.

Sul means on the.

It is a contraction of:

• su = on
• il = the

So:

• su + il = sul

Examples:

• sul tavolo = on the table
• sul pavimento = on the floor

In your sentence, sul tornio means on the wheel.

Because tornio is a singular masculine noun, and Italian normally needs the definite article here.

• il tornio = the wheel / the lathe
• su + il = sul

So sul tornio literally means on the wheel.

Saying su tornio by itself would sound incomplete or unnatural in standard Italian.

Tornio usually means lathe or wheel, depending on context.

By itself, tornio can refer to different kinds of rotating machines. In this sentence, the phrase tornio da vasaio makes the meaning specific: it is a potter's wheel.

So:

• tornio = a wheel/lathe
• tornio da vasaio = potter's wheel

Because da + profession/person can describe what something is used for or what type of thing it is.

So:

• tornio da vasaio = wheel for a potter = potter's wheel

This is very common in Italian:

• scarpe da tennis = tennis shoes
• spazzolino da denti = toothbrush
• camera da letto = bedroom

Here da vasaio tells you the kind of wheel.

Not in this sentence.

Although da can sometimes mean from, here it means something more like for or used by. It describes function or type.

So:

• tornio da vasaio = potter's wheel not
• wheel from the potter

Context is very important with da, because it has several uses in Italian.

They mean different things.

• tornio da vasaio = potter's wheel
  A type of wheel, defined by its use.

• tornio del vasaio = the potter's wheel in the sense of the wheel belonging to the potter

So da vasaio is about category or purpose, while del vasaio is more possessive.

In your sentence, da vasaio is the normal way to name the object itself.

Because in this kind of da + noun expression, Italian usually does not use an article.

So:

• tornio da vasaio
• scarpe da ginnastica
• sala da pranzo

The noun after da functions almost like a label that identifies the type or purpose of the object.

If you added an article, the meaning would usually change.

• argilla is feminine singular
• tornio is masculine singular
• vasaio is masculine singular

You can tell from the articles and forms used:

• l'argilla comes from la argilla, so it is feminine
• sul = su + il, so tornio is masculine
• vasaio is a masculine noun meaning potter

This matters because articles and contractions must agree with the noun.

No, it is very normal Italian word order:

• L'argilla = subject
• è = verb
• sul tornio da vasaio = place expression

So the structure is basically:

Subject + verb + location

Very similar to English:

• The clay is on the potter's wheel.

A simple pronunciation guide is:

Lar-JEEL-la eh sool TOR-nyo da va-ZA-yo

A few notes:

• gli in argilla sounds like the lli in million, but more clearly Italian
• è is pronounced like eh
• gn is not present here, but ni in tornio gives TOR-nyo
• vasaio has three vowel sounds: va-sa-io

A more IPA-style version would be approximately:

/larˈdʒil.la ɛ sul ˈtɔr.njo da vaˈzaːjo/

Yes, grammatically you could, but the meaning shifts slightly.

• sul tornio da vasaio = on the potter's wheel as a type of object
• sul tornio del vasaio = on the potter's wheel meaning the wheel that belongs to the potter

So both can work in some contexts, but tornio da vasaio is the standard name of the object.