Mama obiera ziemniaki, a ja kroję cebulę.

In Polish, family words like mama, tata, babcia, and so on are very often used without a possessive like my when it is already obvious whose family member you mean.

So Mama obiera ziemniaki naturally means Mom is peeling potatoes or My mom is peeling potatoes.

Adding moja is possible, but it often sounds more emphatic or contrastive:

• Moja mama obiera ziemniaki, a twoja gotuje zupę.
  My mom is peeling potatoes, and yours is cooking soup.

A often translates as and, but it is not exactly the same as i.

In this sentence, a connects two contrasting or parallel pieces of information:

• Mama obiera ziemniaki, a ja kroję cebulę.

It feels like:

• Mom is peeling potatoes, while I am cutting the onion.
• Mom peels the potatoes, and I cut the onion.

So a is very commonly used when two subjects are doing different things:

• On czyta, a ona pisze.
  He is reading, and she is writing.

By contrast, i is more of a simple addition:

• Mama i ja gotujemy.
  Mom and I are cooking.

Yes, Polish often drops subject pronouns because the verb ending usually shows who is doing the action.

So you could say:

• Mama obiera ziemniaki, a kroję cebulę.

That is grammatically fine. But ja is included here for clarity or contrast:

• Mama does one thing, I do another.

So a ja sounds natural because it highlights the different subject:

• Mom is peeling potatoes, and I’m cutting the onion.

Because kroję is the 1st person singular present tense form of kroić (to cut / to slice).

Polish verbs change their endings depending on the subject:

• ja kroję — I cut / I am cutting
• ty kroisz — you cut
• on/ona kroi — he/she cuts
• my kroimy — we cut
• wy kroicie — you plural cut
• oni/one kroją — they cut

So:

• infinitive: kroić
• actual sentence form: kroję

The -ę ending is very common for I in the present tense.

Obiera is the present tense form of obierać:

• mama obiera — mom peels / is peeling

Obierze is a future form of the perfective verb obrać:

• mama obierze — mom will peel

So the sentence uses present, ongoing action:

• Mama obiera ziemniaki, a ja kroję cebulę.

That means something like:

• Mom is peeling potatoes, and I’m cutting the onion.

It is accusative plural, because it is the direct object of obiera.

However, for many non-masculine-personal plural nouns, the accusative form is the same as the nominative form.
That is what happens with ziemniaki.

Forms of ziemniak:

• nominative singular: ziemniak
• accusative singular: ziemniaka
• nominative plural: ziemniaki
• accusative plural: ziemniaki

So in this sentence:

• Mama obiera ziemniaki
  Potatoes are the thing being peeled, so they are in the accusative.

Because cebula is the nominative singular form, but after a transitive verb like kroić, the direct object usually goes into the accusative.

So:

• nominative: cebula — onion
• accusative: cebulę — onion as the object

That is why we say:

• kroję cebulę — I am cutting an onion / the onion

This is a very important pattern in Polish:

• Mam książkę. — I have a book.
• Widzę kobietę. — I see a woman.
• Jem zupę. — I eat soup.

Feminine nouns ending in -a often change to -ę in the accusative singular.

Polish does not have articles like a/an and the, so cebulę can mean either:

• the onion
• an onion

The exact meaning depends on context.

So kroję cebulę could mean:

• I’m cutting the onion
  or
• I’m cutting an onion

English forces you to choose an article, but Polish does not.

Both verbs here are imperfective:

• obierać — to be peeling / to peel habitually
• kroić — to be cutting / to cut habitually

Imperfective verbs are used for:

• ongoing actions
• repeated actions
• general descriptions

That fits this sentence perfectly, because it describes what is happening in the kitchen right now.

If you wanted to talk about completing the action, you would often use a perfective verb:

• obrać — to peel completely
• pokroić — to cut up / slice up completely

For example:

• Mama obrała ziemniaki. — Mom peeled the potatoes.
• Pokroiłem cebulę. — I cut up the onion.

Yes. Polish word order is more flexible than English because case endings show grammatical roles.

The neutral order here is:

• Mama obiera ziemniaki, a ja kroję cebulę.

But other orders are possible for emphasis:

• Ziemniaki obiera mama, a cebulę kroję ja.
• Ja kroję cebulę, a mama obiera ziemniaki.

These alternatives are grammatically possible, but they change the focus or emphasis. The original sentence sounds natural and neutral in everyday speech.

A couple of pronunciation points often trouble English speakers:

• kroję: the ję at the end is pronounced roughly like yeh with nasal coloring on the vowel. The j sounds like English y.
• cebulę: the final ę is a nasal vowel. In everyday speech, especially at the end of a word, it is often pronounced less strongly nasal than learners expect.

Also:

• c in cebulę sounds like ts
• j in ja and kroję sounds like y in yes

A rough learner-friendly approximation:

• Mama obiera ziemniaki, a ja kroję cebulę
  ≈ MA-ma o-BYE-ra zhyem-NYA-kee, a ya KRO-ye tseh-BOO-leh

That is only an approximation, but it can help at the beginning.

It can mean either, depending on context.

Polish present tense often covers both:

• I am cutting
• I cut
• I usually cut

So this sentence can mean:

• Mom is peeling potatoes, and I’m cutting the onion.
  if it describes what is happening now

But in another context, similar forms could describe routine:

• Mama obiera ziemniaki, a ja kroję cebulę, kiedy robimy obiad.
  Mom peels potatoes and I cut the onion when we make dinner.

In this sentence, the kitchen context strongly suggests an action happening now.