Danas imam slobodan dan.

Because Serbian often omits subject pronouns when the verb already makes the subject clear.

• imam = I have
• so you do not need ja = I

You can say Ja danas imam slobodan dan if you want extra emphasis on I, but in normal speech Danas imam slobodan dan sounds natural.

Imam is the 1st person singular present tense of imati = to have.

So:

• imam = I have
• imaš = you have
• ima = he/she/it has

In this sentence, imam slobodan dan literally means I have a free day, which is a natural Serbian way to say I have a day off.

Danas means today, and Serbian word order is fairly flexible.

Putting danas first is very natural because it sets the time right away:

• Danas imam slobodan dan. = Today I have a day off.

You could also hear:

• Imam danas slobodan dan.
• Imam slobodan dan danas.

But Danas imam slobodan dan is the most neutral and common order.

Because the adjective must agree with the noun in gender, number, and case.

Here:

• dan = day
• dan is masculine singular
• so the adjective must also be masculine singular

That gives:

• slobodan dan = free day / day off

Other forms would match different noun types:

• slobodna for feminine singular
• slobodno for neuter singular
• slobodni for masculine plural (or sometimes a different function depending on context)

It is in the accusative case, because it is the direct object of imam.

You are literally having something:

• imam what?
• slobodan dan

However, this is a useful Serbian pattern: for masculine inanimate nouns, the accusative often looks the same as the nominative.

So:

• nominative: slobodan dan
• accusative: slobodan dan

That is why nothing visibly changes here.

Yes, it is a normal and understandable expression.

Slobodan basically means free, unoccupied, or not busy, and with dan it gives the idea of a day when you do not work or do not have obligations.

Depending on context, Serbian speakers might also say things like:

• Imam slobodan dan.
• Danas ne radim. = I’m not working today.
• Danas sam slobodan / slobodna. = I’m free today.

So slobodan dan is natural, especially when you mean a day off.

They are similar, but not identical.

• Danas imam slobodan dan = Today I have a day off

  • focuses on the day itself
  • often implies no work / no duties

• Danas sam slobodan (male speaker) / slobodna (female speaker) = I’m free today

  • focuses on your availability
  • can mean you are not busy and can meet, help, go somewhere, etc.

So if you mean I’m not working today, imam slobodan dan is very good.
If you mean I’m available today, sam slobodan/slobodna is often better.

Because Serbian has no articles like English a/an/the.

So slobodan dan can mean:

• a free day
• a day off
• sometimes even the free day, depending on context

Serbian relies on context, word order, and sometimes other words to show whether something is definite or indefinite.

That depends on your gender, but only when the adjective describes you directly.

In your sentence:

• slobodan describes dan (day), not the speaker
• since dan is masculine, it must be slobodan, no matter who is speaking

So both a man and a woman can say:

• Danas imam slobodan dan.

But if you say I am free today, then the adjective matches the speaker:

• male speaker: Danas sam slobodan.
• female speaker: Danas sam slobodna.

A simple approximate pronunciation for an English speaker is:

• DAH-nahs EE-mahm sloh-BOH-dahn dahn

A few helpful notes:

• danas = stress usually on the first syllable: DA-nas
• imam = EE-mam
• slobodan = slo-BO-dan
• dan = like dahn, with a pure a sound, not the English ae in man

Also, Serbian spelling is very regular, so words are usually pronounced much like they are written.

Yes, absolutely.

• Danas imam slobodan dan = neutral, natural
• Ja danas imam slobodan dan = puts extra emphasis on I

For example, you might say ja if you are contrasting yourself with someone else:

• Ti radiš, a ja danas imam slobodan dan.
  = You’re working, and I have a day off today.

So ja is optional here, not wrong.