Ma sœur a peur de glisser sur le carrelage mouillé.

Because avoir peur is the normal French expression for to be afraid.

So:

• Ma sœur a peur = My sister is afraid
• literally, it is closer to My sister has fear

French often uses avoir where English uses to be.
You should learn avoir peur as a fixed expression.

Examples:

• J’ai peur. = I’m afraid.
• Ils ont peur du noir. = They’re afraid of the dark.

After avoir peur, French normally uses de before a verb or thing that causes the fear.

In this sentence:

• a peur de glisser = is afraid of slipping

This is a very common pattern:

• avoir peur de + infinitive
• avoir peur de + noun

Examples:

• Elle a peur de tomber. = She is afraid of falling.
• Il a peur des chiens. = He is afraid of dogs.

So the de is required by the expression avoir peur de.

Because after de, when you talk about an action in a general way, French uses the infinitive.

So:

• avoir peur de glisser = to be afraid of slipping

The infinitive works like English slipping / to slip in this kind of structure.

Other examples:

• Elle essaie de dormir. = She is trying to sleep.
• Nous évitons de parler trop vite. = We avoid speaking too fast.

Here, glisser is not conjugated because it is not the main verb of the sentence. The main verb is a from avoir.

Glisser means to slip or to slide.

In this sentence, the idea is to slip accidentally, especially because the floor is wet.

So:

• glisser sur le carrelage mouillé = to slip on the wet tiled floor

It is close to slip, not exactly fall.
A person can glisser and maybe then tomber.

• glisser = slip
• tomber = fall

Carrelage usually means tiled flooring or tiling.

In this sentence, le carrelage refers to the tiled floor surface, not just one tile.

So:

• un carreau = a tile
• le carrelage = the tiling / tiled floor / tile flooring

That is why sur le carrelage mouillé is best understood as on the wet tiled floor.

French usually needs an article where English sometimes does not.

So French says:

• sur le carrelage mouillé

where English might say:

• on wet tile
• on the wet tiled floor

Using le here is completely normal because French generally uses articles with nouns much more often than English does.

In French, many adjectives come after the noun, and mouillé is one of them.

So:

• le carrelage mouillé = the wet tiled floor

This noun + adjective order is very common:

• une maison blanche = a white house
• un sol glissant = a slippery floor

Some adjectives come before the noun, but mouillé normally comes after it.

Because mouillé agrees with carrelage, not with sœur.

Here:

• carrelage is masculine singular
• so the adjective is mouillé

Agreement patterns:

• masculine singular: mouillé
• feminine singular: mouillée
• masculine plural: mouillés
• feminine plural: mouillées

For example:

• la route mouillée = the wet road
• le sol mouillé = the wet floor

Even though sœur is feminine, mouillé describes the floor, not the sister.

Sœur is pronounced roughly like sur in British English fur, but the exact French vowel is a bit different.

A useful approximation is:

• sœur ≈ suhr

A few things to notice:

• the s is pronounced
• the œu makes one vowel sound
• the final r is the French r

Also, ma sœur is pronounced smoothly together, and in normal speech it sounds something like:

• ma suhr

Literally, it means on the wet tiled floor.

• sur = on

But in context, English naturally understands the cause too: she is afraid of slipping because the tiled floor is wet.

So the French phrase gives the location/surface directly, and the reason is implied.

Yes, but the meaning would change slightly.

• avoir peur de glisser = to be afraid of slipping
• avoir peur de tomber = to be afraid of falling

In this sentence, glisser is more precise because a wet tiled floor especially makes you think of slipping.

Someone might slip and then fall, but the first danger here is glisser.

Because ma sœur already means my sister, so French does not need an extra possessive pronoun before the body or action here.

The sentence simply says what my sister is afraid of:

• Ma sœur = my sister
• a peur de glisser = is afraid of slipping

French does not need to say something like slipping herself. That would sound unnatural here. The meaning is already clear.