Learn Hardcore Polish: Mój dziadek jest spokojny, ale jego brat bywa nerwowy. - My grandfather is calm, but his brother can be nervous.

Lubię mojego dziadka. – I like my grandfather. (direct object → accusative)
Nie ma mojego dziadka. – My grandfather is not here / is missing. (after nie ma, genitive)

What are the forms of mój and why is mój used here?

The possessive mój (my) changes with gender, number and case. In the nominative singular:

mój – with masculine nouns: mój dziadek (my grandfather)
moja – with feminine nouns: moja babcia (my grandmother)
moje – with neuter nouns: moje dziecko (my child)

Since dziadek is a masculine noun in the nominative, you must use mój.

Why is it jego brat, not jego brata?

Jego brat is also the subject of the second clause, so it must be in the nominative:

brat – nominative singular (subject)
brata – would be genitive/accusative, used for objects or after some prepositions/verbs.

Jego here is a possessive pronoun meaning his, and (unlike mój) it does not change its form:

jego brat – his brother
jego siostra – his sister
jego dzieci – his children

In all cases it stays jego; only the noun (like brat/brata) changes for case.

What is the difference between jest and bywa in this sentence?

Both come from the verb być (to be), but they express different nuances:

jest spokojny – is calm → describes a general, stable quality
bywa nerwowy – literally is (from time to time) nervous → means is sometimes / tends to be / can be nervous

So the sentence suggests:

The grandfather is generally calm.
The brother is not always nervous, but from time to time / in some situations he becomes nervous.

That is why Polish uses jest in the first clause and bywa in the second to show the contrast.

Could I say Mój dziadek jest spokojny, ale jego brat jest nerwowy instead?

Yes, it is grammatically correct, but the meaning changes slightly:

bywa nerwowy – is sometimes / tends to be nervous (occasional, not constant)
jest nerwowy – is nervous (sounds like a more permanent characteristic)

With jest nerwowy, you present the brother as simply a nervous person, in a more general way.
With bywa nerwowy, you soften it: he can be nervous, but not all the time.

What exactly does bywa mean, and how is it conjugated?

Bywa is the 3rd person singular of bywać, a frequentative form of być (to be).
It suggests repeated or occasional states.

Conjugation of bywać in the present tense:

ja bywam – I sometimes am / I tend to be
ty bywasz – you sometimes are
on/ona/ono bywa – he/she/it sometimes is
my bywamy – we sometimes are
wy bywacie – you (pl.) sometimes are
oni/one bywają – they sometimes are

In spoken language, bywać is less common than być, but is very useful when you want to say tend to be, occasionally be, etc.

Why are the adjectives spokojny and nerwowy in that exact form?

Both adjectives agree with the gender, number and case of the noun they describe.

dziadek – masculine singular nominative → spokojny
brat – masculine singular nominative → nerwowy

Masculine singular nominative adjectives usually end with -y or -i:

spokojny – calm
nerwowy – nervous
wysoki – tall
ciekawy – interesting

For other genders in nominative singular, you would have:

feminine: spokojna, nerwowa – Moja babcia jest spokojna, ale jej siostra bywa nerwowa.
neuter: spokojne, nerwowe – To dziecko jest spokojne, ale inne bywa nerwowe.

Can I drop jest and just say Mój dziadek spokojny?

In standard Polish, when you describe someone with an adjective, you normally must use jest:

Mój dziadek jest spokojny. – correct, neutral
Mój dziadek spokojny. – sounds incomplete or like a note/headline, not normal full speech

You do see jest omitted in:

headlines: Dziadek spokojny, brat nerwowy
some very informal or telegraphic styles

But for regular sentences, keep jest:
Mój dziadek jest spokojny.

Could I change the word order, like Jego brat bywa nerwowy, ale mój dziadek jest spokojny?

Yes. Polish has relatively flexible word order, especially with simple sentences like this. Both versions are fine:

Mój dziadek jest spokojny, ale jego brat bywa nerwowy.
Jego brat bywa nerwowy, ale mój dziadek jest spokojny.

The choice mostly affects emphasis:

First version emphasizes the calm grandfather first.
Second version emphasizes the nervous brother first.

The basic grammatical roles (subject, verb, predicate) stay clear in both.

Why is the conjunction ale used here, and could I use a instead?

Ale is the most common word for but, and it marks a clear contrast:

Mój dziadek jest spokojny, ale jego brat bywa nerwowy.
→ strong contrast: calm vs nervous.

You could also say:

Mój dziadek jest spokojny, a jego brat bywa nerwowy.

A is weaker; it often means and / while / whereas and can sound a bit more neutral, like simply comparing two people.
In this sentence both ale and a are possible; ale just highlights the contrast more strongly.

What does spokojny really mean? Is it just “calm”?

Spokojny broadly means:

calm, peaceful
quiet, not loud
not easily upset or stressed

Depending on context, it might describe:

a person’s temperament: He is calm, not easily irritated.
a place: spokojna ulica – a quiet street
a situation: spokojna noc – a peaceful night

In this sentence, it describes the grandfather’s character: he is a calm, easy‑going person.

Does nerwowy mean shy, like English “nervous before an exam”?

Not usually. Nerwowy is closer to:

irritable, on edge
easily upset or irritated
tense, high‑strung

So nerwowy often describes someone who gets angry or snappy easily, not someone who is just shy or anxious.

Examples:

On jest bardzo nerwowy. – He is very irritable / high‑strung.
Robi się nerwowy, kiedy się spieszy. – He gets nervous/irritable when he’s in a hurry.

For “nervous before an exam” in the sense of anxious, worried, you might also hear:

zestresowany – stressed
zdenerwowany – upset/nervous in a specific situation

How do you pronounce Mój dziadek jest spokojny, ale jego brat bywa nerwowy?

Approximate pronunciation (using English-like spelling):

Mój – mooy (like “moy” with a slightly longer oo sound)
dziadek – JAH-dek (the dz
- i sound like the j in “jeans”)
jest – yest (like “yes” + t)
spokojny – spo-KOY-nih
- sp as in “spot”
- oj like “oy” in “boy”
- final -ny is one syllable, soft n, short i
ale – AH-leh (both vowels like in “father” and “let”)
jego – YE-go
brat – braht (rolled/strong r, short a)
bywa – BIH-va (short, clear vowels)
nerwowy – ner-VOH-vih

Stress is almost always on the second-to-last syllable:

mój DZIA‑dek jest spo‑KOJ‑ny, A‑le JE‑go brat BY‑wa ner‑WO‑wy.

Why isn’t there any word like “the” or “a” in this Polish sentence?

Polish has no articles (no equivalents of a/an or the).
Whether you mean a grandfather or my/the grandfather is understood from:

context
possessives (like mój, jego)
sometimes word order or emphasis

So:

Mój dziadek jest spokojny – My grandfather is calm.
Dziadek jest spokojny – could be Grandfather is calm (when everyone knows which grandfather you mean), or The/a grandfather is calm, depending on context.

In your sentence, mój and jego already carry the meaning that English would often show with my and his, so no articles are needed.