На столе лежит кусочек сыра.

Because на can mean either onto or on, depending on the case.

• на стол = onto the table → motion toward a surface, so it takes the accusative
• на столе = on the table → location, so it takes the prepositional

In this sentence, the cheese is already located there, not moving there, so Russian uses на столе.

That is the prepositional case singular form of стол.

• dictionary form: стол
• prepositional singular: на столе

Many masculine nouns add -е in the prepositional singular after a preposition of location like на or в.

So:

• стол = table
• на столе = on the table

Russian often prefers a more specific position verb where English would simply say is.

лежит comes from лежать, meaning to lie. It is commonly used for objects that are resting horizontally or simply lying somewhere.

So На столе лежит кусочек сыра is more natural than a literal There is a piece of cheese on the table with just a general verb.

Russian often uses:

• лежать = lie
• стоять = stand
• висеть = hang

depending on how the object is situated.

Because the subject is singular: кусочек.

Even though сыра is also present, it is not the main subject. The core noun phrase is:

• кусочек = little piece

and сыра just tells you of what the piece is.

So the verb agrees with кусочек, which is singular:

• кусочек лежит
• if it were plural: кусочки лежат

Because after nouns meaning a part, amount, or piece, Russian usually puts the second noun in the genitive case.

So:

• кусочек = a small piece
• сыра = of cheese

Together, кусочек сыра literally means a little piece of cheese.

This is very common in Russian:

• кусок хлеба = a piece of bread
• чашка чая = a cup of tea
• бутылка воды = a bottle of water

Сыр is the dictionary form, nominative singular. After кусочек, Russian requires the genitive singular, and for сыр that form is сыра.

So:

• сыр = cheese
• кусочек сыра = a piece of cheese

This -а ending is a very common genitive singular ending for masculine nouns.

Yes. Кусочек is a diminutive form of кусок.

• кусок = a piece, a chunk
• кусочек = a little piece, a small piece

The diminutive can suggest small size, and sometimes a softer or more natural tone. In everyday speech, Russian uses diminutives quite a lot.

So кусочек сыра sounds like a little piece of cheese or a piece of cheese with a slightly more natural, modest feel.

Russian has no articles like English a/an/the.

So кусочек сыра can mean:

• a piece of cheese
• the piece of cheese

Which meaning is intended depends on context.

In this sentence, English would normally translate it as There is a piece of cheese on the table or A piece of cheese is lying on the table, but Russian does not need an article to express that.

Yes, you could. Russian word order is more flexible than English.

Both are grammatical:

• На столе лежит кусочек сыра
• Кусочек сыра лежит на столе

The difference is mostly about focus.

• На столе лежит кусочек сыра puts the location first, like On the table, there is a piece of cheese
• Кусочек сыра лежит на столе starts with the thing itself: A piece of cheese is lying on the table

So the original sentence sounds very natural if you are introducing what is on the table.

Not in the present tense in this exact kind of sentence if you want the idea is lying/is located there in a natural full sentence.

Russian often omits to be in the present tense in sentences like:

• Это сыр. = This is cheese.

But for location/existence, Russian commonly uses a lexical verb such as:

• лежит = is lying
• стоит = is standing
• находится = is located

So На столе кусочек сыра might be understandable in some contexts as a fragment or caption, but as a normal sentence На столе лежит кусочек сыра is the natural form.

It does both.

Russian often uses verbs like лежать not only to describe physical position, but also to introduce the presence of something in a place.

So На столе лежит кусочек сыра can function like:

• There is a piece of cheese on the table
• while also literally meaning
• A piece of cheese is lying on the table

That overlap is very normal in Russian.