Jestli máš hlad, dám ti sendvič se šunkou nebo jogurt.

Here jestli means if.

So Jestli máš hlad... = If you're hungry...

A useful extra point: jestli can also mean whether in other sentences, for example Nevím, jestli přijde = I don't know whether he'll come.

In everyday Czech, jestli is a very common word.

Sometimes, yes, but they are not exactly identical.

• jestli = neutral everyday if
• pokud = more formal if/provided that
• když = usually when, but in colloquial Czech it can sometimes mean if

In your sentence, jestli is the safest and most natural everyday choice.

• Jestli máš hlad, dám ti... = neutral
• Pokud máš hlad, dám ti... = a bit more formal
• Když máš hlad, dám ti... = possible in some contexts, but often sounds more like when(ever) you're hungry

Because Jestli máš hlad is a subordinate clause, and Czech normally separates subordinate clauses with a comma.

So:

• Jestli máš hlad, dám ti sendvič...

This is standard Czech punctuation.

Because mít hlad is the normal Czech expression for to be hungry.

So:

• mám hlad = I'm hungry
• máš hlad = you're hungry
• má hlad = he/she is hungry

English uses be hungry, but Czech usually uses have hunger.

There is also hladový / hladová = hungry, but mít hlad is much more common in everyday speech.

Yes. máš is you have for one person in the informal singular.

So this sentence is addressed to one person you say ty to.

If you wanted the formal or plural version, you would say:

• Jestli máte hlad, dám vám sendvič se šunkou nebo jogurt.

Here:

• máte = you have (formal singular or plural)
• vám = to you

Because dát is a perfective verb.

In Czech, the present-tense forms of perfective verbs usually refer to the future.

So:

• dám literally looks like present tense
• but in meaning it is I will give

Compare:

• dávám = I am giving / I give regularly (imperfective)
• dám = I will give, once / as a completed action (perfective)

So dám ti sendvič naturally means I'll give you a sandwich.

ti means to you.

In dám ti sendvič:

• dám = I will give
• ti = to you
• sendvič = sandwich

So literally: I-will-give to-you sandwich.

Why ti and not tobě?

• ti is the normal unstressed short form
• tobě is the full form, used for emphasis, contrast, or after some prepositions

So:

• Dám ti sendvič. = neutral, normal
• Dám tobě sendvič, ne jemu. = emphasis: I'll give the sandwich to you, not to him

They are both direct objects, so they are in the accusative case.

But both sendvič and jogurt are masculine inanimate nouns, and in the singular their accusative is often the same as the nominative.

So:

• sendvič → accusative sendvič
• jogurt → accusative jogurt

That is why they look unchanged.

se šunkou means with ham.

The preposition is s / se = with, and it takes the instrumental case.

Base noun:

• šunka = ham

Instrumental singular:

• šunkou

So:

• se šunkou = with ham

This is why the ending changes.

s and se are the same preposition. Czech often uses se instead of s when it is easier to pronounce.

Before šunkou, se sounds smoother than s šunkou.

So:

• se šunkou = natural pronunciation

This kind of extra vowel is very common in Czech prepositions.

Because Czech has no articles like a, an, or the.

So Czech simply says:

• dám ti sendvič
• literally: I'll give you sandwich

In natural English, you add the article:

• I'll give you a sandwich
• I'll give you a yogurt

Czech leaves that to context.

Yes. nebo means or, so the sentence offers two alternatives:

• sendvič se šunkou
• jogurt

So the meaning is:

• I'll give you a ham sandwich or a yogurt

Czech often does not repeat words that are understood from context. If you wanted to make it more explicit, you could say:

• Jestli máš hlad, dám ti sendvič se šunkou, nebo ti dám jogurt.

But the original version is completely normal.