Moram da radim ovaj zadatak danas.

Moram means I must or I have to.

It is the 1st person singular present form of morati = to have to / must.

Serbian usually drops subject pronouns when they are already clear from the verb ending. So:

• Moram = I must / I have to
• Ja moram = I must, with extra emphasis on I

So the sentence does not need ja unless you want contrast or emphasis.

After many verbs in Serbian, especially modal verbs like morati, Serbian very often uses:

• da
  • present tense

So:

• moram da radim = I have to do / work

This is one of the most important patterns in Serbian grammar.

In English, you would use an infinitive:

• I have to do

In Serbian, the very common equivalent is:

• moram da radim

So da radim is not a separate tense here; it is the normal structure after moram.

Radim is the 1st person singular present of raditi = to work / to do.

So literally:

• radim = I work / I am doing
• da radim after moram = to do / to work on

In this sentence, radim refers to doing or working on the task.

This is a question about aspect, which is very important in Serbian.

• raditi is imperfective
• uraditi is perfective

Very roughly:

• raditi focuses on the process / activity
• uraditi focuses on completion

So:

• Moram da radim ovaj zadatak danas can suggest I have to work on/do this task today
• Moram da uradim ovaj zadatak danas more strongly suggests I have to get this task done / finish this task today

In real life, if the idea is that the task must be completed today, many speakers might prefer uradim.
If the focus is simply that this is the task you must work on today, radim works.

Because ovaj zadatak is the direct object of the verb radim.

The noun zadatak is:

• masculine
• singular
• inanimate

In Serbian, the accusative singular of masculine inanimate nouns is the same as the nominative singular. So:

• nominative: ovaj zadatak
• accusative: ovaj zadatak

That is why the form does not change here.

Ovaj means this.

It is a demonstrative adjective, and it must agree with the noun in:

• gender
• number
• case

Since zadatak is masculine singular, we get:

• ovaj zadatak = this task

Compare:

• taj zadatak = that task
• onaj zadatak = that task over there / that one

Danas means today.

It tells you when the obligation applies:

• Moram da radim ovaj zadatak danas = I have to do this task today

It is an adverb, so it does not change form here.

Yes, Serbian word order is fairly flexible.

The version you have:

• Moram da radim ovaj zadatak danas

is a very natural, neutral sentence.

But you could also say:

• Danas moram da radim ovaj zadatak
  → puts more emphasis on today

• Ovaj zadatak moram da radim danas
  → puts more emphasis on this task

So the word order can change, but the emphasis changes too.

It can exist, but in Serbian, moram da radim is the more typical and natural pattern.

The infinitive pattern:

• moram raditi

is more associated with styles or varieties where the infinitive is used more often, especially compared with Serbian usage. A Serbian learner should get very comfortable with:

• moram da...
• želim da...
• mogu da...
• treba da...

So for Serbian, Moram da radim ovaj zadatak danas is the safer model to learn first.

Not the basic meaning, but it adds emphasis.

• Moram da radim ovaj zadatak danas = neutral I have to do this task today
• Ja moram da radim ovaj zadatak danas = I have to do this task today

You might add ja if you are contrasting yourself with someone else:

• Ti ne moraš, ali ja moram da radim ovaj zadatak danas.
  = You don’t have to, but I do.