В кафе нам принесли по ломтику батона с ветчиной.

Because в + prepositional case is used to mean in / at a place.

• кафе is an indeclinable noun, so its form does not change.
• So в кафе means in the café or at the café.

Here it sets the scene: In the café, ...

Because нам is the dative case of мы and means to us.

In this sentence, the action is done for / to us:

• мы = we
• нам = to us

So:

• нам принесли = they brought us

Russian often uses the dative for the person receiving something.

Принесли is the past tense plural form of принести, which means to bring.

A few things are going on here:

• при-нести = to bring, to carry and deliver
• принесли = brought
• It is plural past, so the implied subject is they

Russian often leaves out subjects when they are obvious or unimportant. So принесли by itself can mean:

• they brought
• someone brought
• it was brought in a natural English translation, depending on context

Because Russian very often omits the subject if it is understood from the verb form.

In принесли, the ending already tells you the verb is past plural, so a native speaker naturally understands something like:

• they brought
• the staff brought
• the waiter brought

The sentence does not care about who exactly did it; the important part is that the food was brought.

It means a slice each or one slice apiece.

This is a very common Russian pattern:

• по
  • dative often expresses distribution
• it tells you that each person got that amount

So:

• нам принесли по ломтику = they brought us one slice each

This is not just a slice in general; it specifically suggests distribution among the people in нам.

Because after по in this distributive meaning, Russian commonly uses the dative case.

• nominative: ломтик
• dative: ломтику

So:

• по ломтику = one slice each

This is a fixed and very common structure in Russian.

Because ломтик is a noun meaning slice, and the thing being sliced is put in the genitive case.

Compare the logic in English:

• a slice of bread
• a piece of cake

Russian works similarly:

• ломтик батона = a slice of loaf / white bread loaf

So:

• батон = dictionary form
• батона = genitive singular, after ломтик

Батон usually refers to a long white loaf of bread, often somewhat like a baguette-shaped loaf, but not exactly the same as a French baguette.

This can be confusing because English learners may want to translate it simply as loaf or bread. In everyday Russian:

• хлеб = bread in general
• батон = a specific kind of white loaf bread

So ломтик батона is more specific than just a slice of bread. Depending on context, though, English might still translate it naturally as a slice of bread.

Yes, that is the most natural understanding here.

• с ветчиной = with ham
• ветчина = ham
• ветчиной is the instrumental case after с

In this sentence, с ветчиной describes what was brought: a slice of bread with ham.

Russian often places this kind of phrase at the end of the noun phrase, so:

• по ломтику батона с ветчиной

is naturally understood as something like:

• a slice of white bread with ham

not as some special kind of ham loaf.

Because the preposition с meaning with normally requires the instrumental case.

So:

• ветчина = ham
• с ветчиной = with ham

This is one of the first major uses of the instrumental that learners meet.

The sentence is structured like this:

• В кафе = in the café
• нам = to us
• принесли = they brought
• по ломтику батона с ветчиной = a slice each of white bread with ham

So the core is:

• нам принесли ... = they brought us ...

And the full meaning is built by adding:

• location: в кафе
• distributed quantity: по ломтику
• what kind of item: батона с ветчиной

Yes. Russian word order is flexible, and the meaning would stay basically the same.

For example, you could also say:

• Нам в кафе принесли по ломтику батона с ветчиной.
• По ломтику батона с ветчиной нам принесли в кафе.

But the original order sounds natural because it starts with the setting:

• В кафе = in the café

Then it moves to:

• нам = to us
• принесли = brought
• and finally the thing brought

So the original order is a normal, neutral way to present the information.

Because принесли is perfective and presents the action as completed: the slices were brought and delivered.

• принесли = they brought / they have brought
• приносили = they were bringing / used to bring / brought on some occasion, with a different aspectual feel

In a sentence about a completed serving action in a café, принесли is the natural choice.

It focuses on the result:

• now the slices have been brought to us

Grammatically, it is built with singular forms, but the meaning is distributive plural.

• по ломтику literally uses singular
• but it means one slice for each person

So if нам refers to more than one person, the real-life meaning is multiple slices total, even though Russian uses the singular after по.

That is very normal:

• по яблоку = one apple each
• по билету = one ticket each
• по ломтику = one slice each