Vedle banky je bankomat, kde si můžu vzít hotovost.

Because vedle requires the genitive case, and banky is the genitive singular form of banka.

• banka = bank
• vedle banky = next to the bank / beside the bank

This is something English speakers often have to get used to: Czech prepositions often force a specific case.

In normal modern Czech, yes: vedle is followed by the genitive.

Examples:

• vedle domu = next to the house
• vedle školy = next to the school
• vedle banky = next to the bank

So even though the meaning is about location, you do not use the locative here.

Both are possible, but they feel slightly different.

• Vedle banky je bankomat.
  This presents the location first and then introduces what is there. It is very natural when pointing something out: Next to the bank, there is an ATM.
• Bankomat je vedle banky.
  This puts more focus on the ATM itself: The ATM is next to the bank.

Czech word order is more flexible than English, and the first version sounds very natural in this kind of descriptive sentence.

Czech has no articles. There is no direct equivalent of English a/an or the in ordinary sentences.

So:

• banka can mean a bank or the bank
• bankomat can mean an ATM or the ATM

You understand which one is meant from context.

Je is the 3rd person singular of být (to be), so here it means is.

• Vedle banky je bankomat. = There is an ATM next to the bank.

Even though English uses there is, Czech simply uses is:

• je bankomat = literally is ATM, but naturally translated as there is an ATM

Here kde means where and introduces a relative clause:

• bankomat, kde si můžu vzít hotovost
• an ATM where I can get cash

It refers back to bankomat.

English learners sometimes expect something more like ve kterém (in which), but kde is very common and natural with places.

Si is the unstressed dative form of the reflexive pronoun se/si. In this sentence, it adds the sense of doing something for oneself.

So:

• vzít si hotovost = literally something like take oneself cash
• naturally: take/get cash, withdraw cash

In many everyday expressions, si does not translate directly into English, but it is still important in Czech because it makes the phrase sound natural and idiomatic.

Sometimes Czech allows that, but here si sounds more natural.

• můžu si vzít hotovost = natural, idiomatic
• můžu vzít hotovost = understandable, but less natural in this context

At an ATM, Czech often uses verbs like:

• vybrat si hotovost
• vybrat peníze
• vzít si hotovost

So si is a very normal part of the expression.

Both mean I can.

• můžu = very common in everyday speech
• mohu = a bit more formal or careful

So:

• můžu is the normal conversational form
• mohu is also correct

A learner should recognize both.

Because Czech usually omits subject pronouns when they are already clear from the verb form.

• můžu already means I can
• so já is not necessary

You can say já můžu if you want extra emphasis, contrast, or clarity, but the neutral sentence normally leaves já out.

Vzít is a perfective verb, while brát is imperfective.

Very roughly:

• brát = to be taking / to take in an ongoing or repeated sense
• vzít = to take as a complete act

After můžu, Czech often uses a perfective verb when talking about one complete possible action:

• můžu vzít = I can take / get

That fits well with an ATM: one complete act of getting cash.

Often, yes. In real-life Czech, people very commonly say:

• vybrat peníze z bankomatu = withdraw money from an ATM
• vybrat si hotovost = withdraw/get cash

But vzít si hotovost is still understandable and works in context. It sounds a little more like take/get cash than the more specific withdraw cash.

Here hotovost is the direct object of vzít, so it is in the accusative case.

For the noun hotovost, the nominative and accusative singular have the same form:

• nominative: hotovost
• accusative: hotovost

So even though the form does not change, the function does: here it is the thing being taken.

Hotovost means cash.

It refers to money in physical form, not money in a bank account. In the context of an ATM, it means the cash you withdraw.

A related word:

• peníze = money

So:

• vybrat peníze = withdraw money
• vzít hotovost = get/take cash

Bankomat is a masculine inanimate noun.

You can see that in forms like:

• nominative singular: bankomat
• accusative singular: bankomat (same form, because it is masculine inanimate)

In your sentence, bankomat is in the nominative because it is the subject of je.

A useful breakdown is:

• Vedle banky = next to the bank
• je bankomat = there is an ATM
• kde = where
• si můžu vzít hotovost = I can get cash

So the sentence is built as:

[location] + [there is X] + [relative clause describing X]

That is a very common Czech pattern.