Moja sestra kaže da je umorna.

Because sestra is a feminine singular noun, and the possessive word moj has to agree with it.

So:

• moj = masculine singular
• moja = feminine singular
• moje = neuter singular

Since sestra is feminine, you say moja sestra.

For the same reason: it agrees with sestra.

The adjective umoran means tired, but its form changes depending on gender:

• umoran = masculine
• umorna = feminine
• umorno = neuter

So because the person being described is sestra, you use umorna.

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

So sestra can mean:

• a sister
• the sister
• just sister

The exact meaning depends on context. In this sentence, moja sestra naturally means my sister, and no article is needed.

It is in the nominative singular, because it is the subject of the main clause.

The main clause is:

• Moja sestra kaže = My sister says

Since my sister is the one doing the action of saying, Serbian uses the nominative form sestra.

Kaže is the 3rd person singular present form of the verb kazati / reći-type reporting meaning to say.

Here it means:

• says
• sometimes, depending on context, is saying

So Moja sestra kaže... = My sister says...

Serbian present tense often covers what English expresses with either the simple present or present progressive, depending on context.

Here da introduces a clause that works like English that.

So:

• Moja sestra kaže da je umorna.
• My sister says that she is tired.

In everyday English, that is often omitted, but in Serbian da is normally kept in sentences like this.

Because je is the verb is.

The part je umorna literally means:

• is tired

So the subordinate clause is:

• da je umorna = that she is tired

Even though Serbian often drops subject pronouns, it still needs the verb. So you can omit ona (she), but you cannot omit je here.

In Serbian, subject pronouns are often omitted when the meaning is already clear from the verb form or context.

So:

• da je umorna = that she is tired
• da je ona umorna also works, but it sounds more emphatic

You would use ona if you want contrast or emphasis, for example:

• Moja sestra kaže da je ona umorna, a ne brat.
• My sister says that she is tired, not my brother.

Without emphasis, leaving out ona is the normal choice.

Because je is a clitic in Serbian, and clitics usually go near the beginning of the clause.

After da, the natural order is:

• da je umorna

This is the standard and expected order.

So even though English puts tired after is, Serbian is also doing that here, but an important extra point is that je has a special placement pattern as a clitic.

Not always.

In this sentence, yes:

• da = that

But Serbian da is also used in other structures where English would not use that, for example:

• Hoću da idem. = I want to go.
  • literally closer to I want that I go, but that is not how English says it

So in Moja sestra kaže da je umorna, da is best understood as that, but in other sentences it may have a different function.

Yes. That is usually the most natural English translation.

English often omits that:

• My sister says that she is tired.
• My sister says she’s tired.

Both match the Serbian sentence. Serbian, however, normally keeps da in this kind of construction.

It is used for a feminine noun or feminine person, not simply for biological sex in isolation.

Here it is feminine because of sestra, which is grammatically feminine.

Compare:

• Brat je umoran. = Brother is tired.
• Sestra je umorna. = Sister is tired.

So the adjective form follows grammatical gender.

Yes, grammatically you can.

Adding moja makes it specifically my sister. Without it, sestra means just sister / the sister, depending on context.

So:

• Moja sestra kaže da je umorna. = My sister says she is tired.
• Sestra kaže da je umorna. = Sister / The sister says she is tired.

In real usage, whether that sounds natural depends on the context.