Дети катают мяч по полу.

Дети is the nominative plural form, because it is the subject of the sentence: the children are the ones doing the action.

A useful thing to know is that дети is an irregular plural. It corresponds to ребёнок (child) in the singular.

• ребёнок = child
• дети = children

So in this sentence, дети means children and is in the form you expect for the subject.

Катают comes from the infinitive катать.

Here it is in the present tense, 3rd person plural, because the subject is дети (they).

A quick breakdown:

• я катаю = I roll
• ты катаешь = you roll
• он/она катает = he/she rolls
• мы катаем = we roll
• вы катаете = you roll
• они катают = they roll

Since дети is plural, Russian uses the they form: катают.

Because катают is a verb used with a direct object, and here that object is мяч (ball).

• катать что? = to roll something
• кататься = to ride / to go for a ride / to roll oneself around

So:

• Дети катают мяч = The children are rolling a ball.
• Дети катаются = The children are riding / having a ride / rolling around themselves.

The -ся ending changes the meaning. Without -ся, the action is done to something. With -ся, the subject is more involved in the action itself.

It is in the accusative. It just happens that for an inanimate masculine noun like мяч, the accusative singular looks exactly the same as the nominative singular.

So:

• мяч = nominative singular
• мяч = accusative singular

Because the ball is the direct object of катают, it needs the accusative, but the form does not visibly change.

This is very common in Russian:

• Я вижу стол.
• Он катает мяч.

In both cases, the object is masculine singular and inanimate, so the accusative looks like the dictionary form.

Because the preposition по here requires the dative case.

The noun is:

• пол = floor

Its dative singular form is:

• полу

So:

• по полу = along the floor / across the floor / around on the floor

This is a very common pattern: по + dative for movement over, along, or around a surface or area.

Not quite.

• на полу means on the floor as a location
• по полу suggests movement along / across / around the floor

So:

• Мяч на полу. = The ball is on the floor.
  (location)
• Дети катают мяч по полу. = The children are rolling the ball along/across the floor.
  (movement over the floor)

That is why по полу is the natural choice here: the sentence describes motion, not just location.

It can mean either, depending on context.

Russian present tense often covers both:

• They are rolling a ball
• They roll a ball

In this sentence, most learners will naturally understand it as are rolling, because it describes an action happening in the scene.

So катают is present tense imperfective, and in English it may be translated either as:

• The children are rolling a ball along the floor
• The children roll a ball along the floor

The first is usually more natural if you are describing what is happening right now.

This is a very good question, because both verbs can relate to rolling.

Very roughly:

• катать often suggests rolling something around, back and forth, or in a less goal-directed way
• катить more often suggests moving something by rolling it in a more directed way, often toward some destination

With a ball in a casual play context, катать мяч is very natural.

So Дети катают мяч по полу gives the feeling of the children rolling the ball around on the floor.

A learner does not need to master the full distinction immediately, but it is useful to feel that катать often fits playful, repeated, or non-one-directional rolling.

The sentence uses a very neutral word order:

• Дети катают мяч по полу.

This is basically subject + verb + object + prepositional phrase, which is often the safest order for learners.

But Russian word order is flexible, and different orders can shift emphasis. For example:

• Мяч дети катают по полу.
  Emphasis on the ball
• По полу дети катают мяч.
  Emphasis on along the floor
• Дети по полу катают мяч.
  Also possible, with slightly different rhythm/emphasis

So the original sentence is neutral and natural, but not the only possible order.

The normal stress is:

• Де́ти ката́ют мяч по по́лу.

A rough pronunciation guide:

• Де́ти = DYE-ti
• ката́ют = ka-TA-yut
• мяч = roughly myach
• по по́лу = pa PO-lu

A few helpful notes:

• я in мяч gives a ya-type sound
• the unstressed о in по is pronounced closer to a in normal speech
• the stress in по́лу is on the first syllable

Because Russian has no articles.

So Russian does not normally have separate words for:

• a
• an
• the

That means мяч can mean:

• a ball
• the ball

The exact meaning depends on context.

So Дети катают мяч по полу could be understood as:

• The children are rolling a ball along the floor
• The children are rolling the ball along the floor

English forces you to choose an article, but Russian usually does not.