Quidam puer magistram “tristem” appellat, sed ego eam iustam appello.

Quidam means something like a certain or some.

So quidam puer = a certain boy / some boy.

It is an indefinite word: it introduces someone without naming him specifically.
It agrees with puer, so it is masculine, singular, and nominative here.

Because they have different jobs in the sentence.

• puer is the subject, so it is in the nominative case.
• magistram is the direct object, so it is in the accusative case.

So:

• puer = the boy as the one doing the action
• magistram = the teacher as the person being called something

This is one of the most important things in Latin: endings show function.

Because with appellare meaning to call, Latin often uses a double accusative pattern:

• the person being called = accusative
• the name/description used for that person = accusative

So:

• magistram tristem appellat = he calls the teacher sad
• eam iustam appello = I call her just/fair

In English, we do not usually notice this as a special construction, but in Latin both parts go into the accusative.

Both adjectives agree with the noun or pronoun they describe, but they belong to different adjective patterns.

They both describe magistram / eam, which are:

• feminine
• singular
• accusative

So each adjective must also be feminine singular accusative.

But their dictionary forms are different:

• tristis, triste is a third-declension adjective
  
  • feminine accusative singular = tristem
• iustus, -a, -um is a first/second-declension adjective
  
  • feminine accusative singular = iustam

So the forms are different because the adjectives decline differently, even though they are matching the same case, number, and gender.

Because the subject changes.

• appellat = he/she calls
  This matches quidam puer = a certain boy
• appello = I call
  This matches ego = I

So the verb ending tells you who is doing the action:

• -t = third person singular
• -o = first person singular

This is why Latin often does not need subject pronouns at all.

It is included for emphasis or contrast.

Latin could have said simply:

sed eam iustam appello

and that would still mean but I call her just/fair.

But sed ego gives a stronger contrast:

• the boy says one thing
• I say something different

So ego is not necessary for basic grammar, but it is very natural when the speaker wants to stress I.

Because eam means her, and it refers back to magistram.

This avoids unnecessary repetition.

• magistram = the teacher
• eam = her

eam is the feminine singular accusative form of is, ea, id.

It is accusative because it is still the direct object of appello.

Eam is:

• accusative
• singular
• feminine

It refers to magistram, so it must match that antecedent in gender and number.
Its case comes from its function in the sentence: it is the direct object of appello.

The dictionary form is magistra.

• magistra = nominative singular
• magistram = accusative singular

So in a vocabulary list you would normally learn magistra, -ae, meaning female teacher or mistress depending on context.

It can mean just, fair, upright, or reasonable, depending on context.

In this sentence, a natural English meaning is probably fair or just.

So the contrast is something like:

• the boy calls her sad
• I call her fair/just

The exact English word depends on the translation style, but the Latin idea is that the speaker gives her a positive moral description.

That usually signals emphasis on the actual label being used.

It suggests something like:

• the boy calls the teacher sad
• perhaps that is the word he uses
• or the writer wants to highlight it as a characterization

This is more a matter of punctuation or presentation than Latin grammar itself.
The grammar would work the same without any special marking:

Quidam puer magistram tristem appellat...

Quite free. Latin relies much more on endings than on word order.

This sentence is arranged clearly and naturally, but many rearrangements would still mean basically the same thing, because the endings show the roles:

• puer = subject
• magistram / eam = object
• tristem / iustam = accusative complement
• appellat / appello = verb

For example, Latin could move words around for emphasis.
But the given order is straightforward and easy to read.

Yes. Magistra is feminine.

So magistram means the female teacher.

If the teacher were male, Latin would use magister and in the accusative magistrum.

That is why the pronoun is eam and the adjective is iustam: they are all feminine.

Yes, especially in the second clause.

Latin does not repeat magistram, because eam is enough.
It also does not need a separate word for is in either clause, because the verb forms already contain that idea:

• appellat = he calls
• appello = I call

So compared with English, Latin often packs more information into endings and uses fewer separate words.