La arcilla tiene una burbuja de aire.

La is the feminine singular definite article, meaning the.

It appears because arcilla is a feminine singular noun: la arcilla = the clay.

In Spanish, nouns usually need an article more often than in English, especially when talking about a specific thing or substance in a sentence like this.

Yes. Arcilla is a feminine noun, so it takes feminine words with it:

• la arcilla
• una arcilla
• arcilla roja

This is something you simply learn with the noun. There is no special ending here like -a that guarantees it in every case, but in this case the noun is indeed feminine.

Tiene comes from tener = to have.

Here, the subject is la arcilla, so the sentence literally says that the clay has an air bubble.

Spanish often uses tener in places where English also uses have.
You could also express a similar idea with hay:

• La arcilla tiene una burbuja de aire.
• Hay una burbuja de aire en la arcilla.

These are both natural, but they are structured differently:

• tiene = the clay has
• hay = there is

So the original sentence focuses on the clay as the subject.

Because the subject is la arcilla, which is third person singular.

The verb tener changes depending on the subject:

• yo tengo
• tú tienes
• él / ella / usted tiene
• nosotros tenemos
• vosotros tenéis
• ellos / ellas / ustedes tienen

Since la arcilla is singular, you use tiene.

This is a very common Spanish pattern:

noun + de + noun

Here, burbuja de aire means an air bubble, literally bubble of air.

Spanish often uses de where English uses a noun acting like an adjective:

• burbuja de aire = air bubble
• vaso de agua = glass of water
• zapatos de cuero = leather shoes

So de aire tells you what kind of bubble it is.

Because de aire is being used as a descriptive phrase: a bubble of air.

Spanish often omits the article after de when the second noun is being used in a general, classificatory way:

• una taza de café
• un vaso de agua
• una burbuja de aire

If you said de un aire or del aire, that would usually change the meaning and sound unnatural here.

Aire is normally a masculine noun: el aire.

For example:

• el aire frío
• un aire extraño

In the sentence de aire, you do not see the article, so the gender is less obvious, but the noun itself is masculine.

In standard Peninsular Spanish, arcilla is usually pronounced approximately as:

ar-THI-ya
or in some areas more like
ar-SEE-ya

A few details:

• ci in most of Spain is pronounced like th in think
• ll is usually pronounced like the y sound in yes in most modern varieties of Spain

So a broad Spain pronunciation would be close to ar-THI-ya.

Yes, but the original order is the most straightforward.

You could also say:

• Hay una burbuja de aire en la arcilla.

That means nearly the same thing, but the focus changes:

• La arcilla tiene... focuses on the clay
• Hay... en la arcilla focuses on the existence of the bubble

Both are natural, but they are not identical in emphasis.

Normally, no. In standard Spanish, you would usually need the article here:

• La arcilla tiene una burbuja de aire.

If there is more than one, you could say:

• La arcilla tiene burbujas de aire.

So:

• una burbuja = one air bubble
• burbujas = air bubbles in general or several of them

Leaving out una in the singular would sound incomplete or unnatural in this sentence.