После ужина я убираю тарелки со стола.

Because после requires the genitive case.

• Dictionary form: ужин = dinner/supper
• Genitive singular: ужина

So:

• после ужина = after dinner

This is something you simply learn with the preposition:

• до урока = before the lesson
• после работы = after work
• после ужина = after dinner

Убираю is the 1st person singular present tense of the imperfective verb убирать.

• убирать = to clear away, tidy up, remove, clean up
• убираю = I clear away / I am clearing away / I usually clear away

Here the imperfective makes sense because the sentence most naturally describes a habitual action:

• После ужина я убираю тарелки со стола.
  = After dinner, I clear the plates from the table.

If you wanted a single completed future action, Russian would usually use the perfective verb:

• После ужина я уберу тарелки со стола.
  = After dinner, I will clear the plates from the table.

Most likely it means a habitual action: I clear the plates from the table after dinner.

In Russian, the present tense of an imperfective verb can mean:

• something happening right now
• something happening regularly / habitually

In this sentence, because of После ужина and the general context, English usually understands it as a routine:

• After dinner, I clear the plates from the table.

If the situation were describing what is happening at the moment, it could also be understood as:

• I’m clearing the plates from the table after dinner.

It is in the accusative plural, but for inanimate plural nouns, the accusative usually looks the same as the nominative.

• Nominative plural: тарелки = plates
• Accusative plural: тарелки = plates

That is why the form does not visibly change here.

Compare:

• Я вижу тарелки. = I see the plates.
• Тарелки стоят на столе. = The plates are on the table.

Same form, different function.

Because the sentence expresses movement from the surface of the table, and Russian uses:

• с / со + genitive = from

So:

• стол = table
• стола = genitive singular
• со стола = from the table

This is different from location:

• на столе = on the table

So compare:

• Тарелки на столе. = The plates are on the table.
• Я убираю тарелки со стола. = I remove the plates from the table.

Со is a pronunciation-friendly variant of с.

Russian often uses со instead of с when the next word begins with a difficult consonant cluster, especially something like ст-, ск-, сш-, and similar combinations.

So:

• с братом
• but со стола

It is mainly about ease of pronunciation.

Стола is the genitive singular of стол.

That happens because the preposition с / со in the meaning from takes the genitive case.

• стол = table
• со стола = from the table

This is a very common pattern:

• из дома = from the house
• с полки = from the shelf
• со стола = from the table

Yes. Very often, Russian omits subject pronouns when the verb ending already makes the subject clear.

So both are possible:

• После ужина я убираю тарелки со стола.
• После ужина убираю тарелки со стола.

Since убираю clearly means I clear away, я is not strictly necessary.

Including я can add:

• slight emphasis
• clarity
• contrast

For example:

• После ужина я убираю тарелки, а он моет посуду.
  = After dinner, I clear the plates, and he washes the dishes.

No, Russian word order is fairly flexible, though some orders sound more natural than others depending on emphasis.

The given sentence is neutral and natural:

• После ужина я убираю тарелки со стола.

You could also hear:

• Я убираю тарелки со стола после ужина.
• После ужина убираю тарелки со стола.

The meaning stays basically the same, but the focus shifts a little.

• Starting with После ужина highlights the time.
• Starting with я can make the subject more prominent.
• Putting после ужина later can sound a bit more matter-of-fact in some contexts.

It can mean several related things, depending on context.

With тарелки со стола, it means something like:

• clear away
• remove
• take away

So here it is not exactly to clean the table. It is specifically to take the plates off the table.

That is very common Russian usage:

• убирать игрушки = put the toys away
• убирать комнату = clean/tidy the room
• убирать тарелки со стола = clear the plates from the table

So the core idea is putting something away / removing it from where it is.

Russian has no articles.

So тарелки can mean:

• plates
• the plates
• some plates

and стола can mean:

• a table
• the table

The exact meaning depends on context.

In this sentence, English naturally uses the:

• After dinner, I clear the plates from the table.

But Russian does not need a separate word for that.

Yes. Ужин is often translated as dinner or supper, depending on the variety of English and the meal system being used.

So после ужина could be understood as:

• after dinner
• after supper

That is a translation choice, not a grammar difference. The Russian structure stays the same.