L’atelier de poterie contient le four de potier.

Because atelier starts with a vowel sound. In French, le and la usually shorten before a vowel or mute h:

• le atelier → l’atelier
• la école → l’école

This is called elision.
Also, atelier is a masculine noun, so the full form would be le atelier, but French does not keep it that way before a vowel.

Atelier means workshop or studio, especially a place where someone makes things by hand or does artistic work.

So:

• un atelier = a workshop / studio
• un atelier de poterie = a pottery workshop / pottery studio

English has also borrowed atelier, but in French it is a normal everyday word.

De poterie means for pottery or related to pottery.

In French, noun + de + noun is very common for naming a type of place or object:

• atelier de poterie = pottery workshop
• musée d’art = art museum
• salle de bain = bathroom

Here, poterie means pottery. So atelier de poterie is not literally just workshop of pottery in an ownership sense; it means a workshop used for pottery.

They are different kinds of words:

• poterie = pottery
• potier = potter

So in the sentence:

• atelier de poterie = pottery workshop
• four de potier = potter’s kiln / pottery kiln

A useful way to remember it:

• -erie often gives a thing, activity, or trade-related place/category
• potier is the person who makes pottery

The feminine form of potier is potière.

This is an important distinction.

• four de potier = a potter’s kiln as a type of kiln
• four du potier = the potter’s kiln, meaning the kiln belonging to a specific potter

So:

• de potier describes the kind of kiln
• du potier shows possession by one particular potter

In your sentence, le four de potier is understood as the potter’s kiln / the pottery kiln, not the kiln of the potter.

Because after one noun, French often uses plain de to classify or describe another noun.

Examples:

• atelier de poterie
• machine à laver
• verre à vin
• salle de classe

This structure often works like an English compound noun. It is usually about type, function, or category, not ownership.

So:

• atelier de poterie = a workshop for pottery
• four de potier = a kiln used by/associated with potters

Le four means the kiln, while un four would mean a kiln.

French often uses the definite article more readily than English, especially when the speaker has a specific item in mind or is talking about the expected item in a certain place.

So:

• L’atelier de poterie contient le four de potier. = The pottery workshop contains the potter’s kiln.
• L’atelier de poterie contient un four de potier. = The pottery workshop contains a potter’s kiln.

Both are grammatically possible, but they do not mean exactly the same thing.

Yes, grammatically it is fine. Contient is the third person singular of contenir, meaning to contain.

So:

• L’atelier ... contient ... = The workshop contains ...

That said, in everyday French, depending on context, people might also say:

• Il y a un four de potier dans l’atelier de poterie. = There is a potter’s kiln in the pottery workshop.
• L’atelier de poterie abrite un four de potier. = The pottery workshop houses a potter’s kiln.

So contient is correct, but it can sound a little more formal or descriptive than il y a.

A careful pronunciation is approximately:

Lat-uh-lyay duh po-tuh-ree con-tyan luh foor duh po-tyay

A more precise IPA version is:

/latəlje də pɔtʁi kɔ̃tjɛ̃ lə fuʁ də pɔtje/

A few pronunciation notes:

• atelier sounds roughly like a-tuh-lyay
• contient has a nasal vowel in -tient
• four sounds like foor, but with a French r
• potier ends like tyay

The l’ links directly into atelier:
L’atelier is pronounced smoothly, not as two separate words.

Here are the genders:

• atelier — masculine
• poterie — feminine
• four — masculine
• potier — masculine

You can see this in the articles:

• l’atelier = from le atelier → masculine
• le four = masculine

Even though poterie is feminine, there is no article before it here because it comes after de in a descriptive expression: atelier de poterie.

Yes. In French, four does not mean the number 4. It means oven or kiln.

Examples:

• un four = an oven / a kiln
• un four de potier = a potter’s kiln

The number four in English is completely unrelated in meaning, even though it is spelled the same.

A basic plural version would be:

Les ateliers de poterie contiennent les fours de potier.

That means The pottery workshops contain the potters’ kilns or the pottery kilns, depending on context.

Changes:

• L’atelier → Les ateliers
• contient → contiennent
• le four → les fours

Notice that de poterie and de potier stay the same in form here.