Learn Hardcore Czech: Můj bratr teď není ve škole, je doma. - My brother is not at school now, he is at home.

Můj means my. It’s the masculine singular form of the possessive pronoun for I.
Czech changes the form depending on the gender and case of the noun:

můj bratr = my (male) brother – masculine singular
moje sestra = my sister – feminine singular
moje auto = my car – neuter singular

Here bratr is masculine, so you use můj.

Why is it Můj bratr and not just Bratr for my brother?

Czech usually expresses possession explicitly with a possessive word like můj (my), tvůj (your), jeho (his), etc.
So Bratr teď není ve škole means The brother is not at school now (or just Brother isn’t at school, but without saying whose).
If you want to say my brother, you normally add můj: Můj bratr.

Where does teď (now) usually go in a Czech sentence? Could I move it?

In this sentence, teď comes after the subject phrase: Můj bratr teď není ve škole.
You can move it for emphasis or style:

Teď můj bratr není ve škole, je doma. – Now my brother is not at school, he’s at home.
Můj bratr není teď ve škole, je doma. – My brother is not at school now, he’s at home.

All are grammatically correct; the differences are mostly in emphasis and rhythm.

Why is it není and not something like ne je for is not?

Czech usually forms the present tense negative of být (to be) as a single word:

je → není (is → is not)
jsem → nejsem (I am → I am not)
jsi → nejsi / nejsi (you are → you are not)

So you say Můj bratr není ve škole = My brother is not at school, not ne je.

What does ve škole literally mean, and what case is škole?

ve škole literally means in (the) school / at school.
The preposition v/ve usually takes the locative case when it means in/at (a place).
The basic form is škola (nominative), but after v/ve (in this meaning) it becomes ve škole (locative).

Why is it ve škole and not v škole?

Czech has both v and ve as forms of the same preposition in/at.
You use ve mainly:

before some words that start with certain consonant clusters, or
when v alone would be hard to pronounce or sound awkward.

v škole is awkward to pronounce for Czech speakers, so it becomes ve škole. Similar examples: ve městě, ve vodě, ve škole.

Why is there a comma before je doma?

The sentence has two clauses joined together:

Můj bratr teď není ve škole – My brother is not at school now
je doma – (he) is at home

In Czech, you usually separate such coordinated clauses with a comma, even if there is no conjunction like a (and). So you get:
Můj bratr teď není ve škole, je doma.

Why is there no on (he) in the second part je doma?

Czech is a pro-drop language: if the subject is clear from context, you can (and usually do) omit the pronoun.
We already know the subject is můj bratr, so repeating on would sound unnecessary:

Natural: Můj bratr teď není ve škole, je doma.
Over-explicit: Můj bratr teď není ve škole, on je doma. (understood, but usually only for strong emphasis)

The subject is understood as he from the context.

What exactly is the difference between doma and something like v domě?

Both relate to being at a house, but they’re used differently:

doma = at home (general home, where you live)
- Je doma. – He is at home.
v domě = in the house (building) – more literal, focusing on the physical building
- Je v domě. – He is in the house (not outside).

In this sentence, je doma is the natural way to say he is at home (as opposed to at school).

Could I change the word order, for example Teď není můj bratr ve škole, je doma? Does it change the meaning?

Yes, you can change the word order, and the basic meaning stays the same, but the emphasis shifts. For example:

Můj bratr teď není ve škole, je doma. – neutral; just stating the fact.
Teď není můj bratr ve škole, je doma. – emphasizes now and my brother; maybe contrasting with some other time or person.
Můj bratr není ve škole teď, je doma. – slightly unusual, putting focus on now, often contrastive.

Czech word order is flexible; you mainly use it to express what is new information and what is emphasized.