U banci je danas veliki red.

Because u means in or at here, and when it shows a fixed location in Serbian, the noun usually goes into the locative case.

So:

• banka = bank
• u banci = in / at the bank

You use u + locative for being somewhere, not moving somewhere.

Compare:

• u banci = in/at the bank
• u banku = into the bank (movement toward it)

Banci is locative singular.

The base noun is:

• banka = bank

In the locative singular:

• banka → banci

This is very common after u when talking about location.

This is a normal sound change in Serbian declension.

The noun ends in -ka, and before the ending -i, the k changes to c:

• banka
• banci

You will see the same kind of pattern in other words too. It is something learners usually just get used to through exposure.

Je is the 3rd person singular present of biti (to be).

Here it works like the verb in an English sentence such as:

• There is a big line at the bank today.

Serbian often uses je in this kind of sentence where English uses there is.
So even though English has there is, Serbian does not need a separate word for there in this structure.

The subject is veliki red.

More specifically:

• red is the noun
• veliki describes it

You can tell red is the subject because it is in the nominative singular, and the verb je is also singular.

The other parts are not the subject:

• u banci = location
• danas = time

Because red is a masculine noun, so the adjective must match it.

Agreement in Serbian means the adjective changes to match the noun in:

• gender
• number
• case

So here:

• red = masculine, singular, nominative
• therefore veliki = masculine, singular, nominative

That is why you get:

• veliki red

not:

• velika red

In this sentence, red means line or queue.

This word can have several meanings depending on context, such as:

• line / queue
• order
• row
• turn

In U banci je danas veliki red, the bank context makes queue/line the natural meaning.

Because Serbian word order is fairly flexible.

Danas means today, and adverbs of time can often move around without changing the basic meaning.

For example, these are all possible:

• U banci je danas veliki red.
• Danas je u banci veliki red.
• U banci danas je veliki red.

They all mean basically the same thing, but the emphasis shifts slightly. The original sentence sounds natural and neutral.

In Serbian, the end of the sentence often carries the newest or most important information.

So in:

• U banci je danas veliki red.

the sentence builds up like this:

• location: u banci
• time: danas
• key information: veliki red

Ending with veliki red makes that the main point: the important thing is that there is a big queue.

It can correspond to either in the bank or at the bank in English, depending on context.

Serbian uses u banci naturally for the location, while English may choose:

• in the bank
• at the bank

So even if the Serbian literally looks like in the bank, the best English translation may sometimes be at the bank.

Because Serbian does not have articles like English a/an/the.

So:

• veliki red could mean a big line
• or, depending on context, the big line

Usually the context tells you which English article makes most sense.

That is a very common difference between English and Serbian.

Yes. Serbian allows several word orders, for example:

• Danas je u banci veliki red.
• Veliki red je danas u banci.
• U banci danas je veliki red.

These are all understandable, but they do not all sound equally natural in every context.

The original:

• U banci je danas veliki red.

is a very natural, standard way to say it. It gives the sentence a smooth flow and puts the focus on veliki red at the end.