Вчера мама сварила густой суп с фасолью и зелёным горохом.

Because сварила is the perfective form, from сварить. It presents the cooking as a completed result: the soup got cooked.

• сварила = cooked, made, boiled and finished
• варила = was cooking, used to cook, cooked in an ongoing/process sense

In this sentence, the speaker is talking about a completed event that happened yesterday, so сварила is the natural choice.

In the past tense, Russian verbs agree with the subject in gender and number.

The subject is мама, which is feminine singular, so the verb must also be feminine singular:

• masculine: сварил
• feminine: сварила
• neuter: сварило
• plural: сварили

So мама сварила means Mom cooked.

Here суп is the direct object of сварила. Direct objects often go into the accusative case.

But суп is:

• masculine
• singular
• inanimate

For masculine inanimate nouns, the accusative singular looks exactly like the nominative singular.

So:

• nominative: густой суп
• accusative: густой суп

That is why the form does not visibly change.

Because after с meaning with, Russian normally uses the instrumental case.

So:

• с фасолью = with beans
• с зелёным горохом = with green peas

The instrumental forms here are:

• фасоль → фасолью
• зелёный горох → зелёным горохом

In food descriptions, с + instrumental is very common for ingredients or things included in a dish.

Because зелёным has to agree with горохом in:

• gender: masculine
• number: singular
• case: instrumental

The basic adjective is зелёный. In the instrumental masculine singular, it becomes зелёным.

So:

• nominative: зелёный горох
• instrumental: с зелёным горохом

This is ordinary adjective agreement.

Russian often uses горох as a singular noun where English uses a plural idea like peas.

So зелёный горох can mean:

• green peas
• green pea

depending on context, but in a food sentence it usually means green peas as an ingredient.

This is just one of those places where Russian and English package the idea differently.

Мама is in the nominative case because it is the subject of the sentence, the person doing the action.

Even though мама ends in -а, it is not an object here. It is simply the feminine singular subject:

• мама сварила = Mom cooked

No. Вчера is an adverb meaning yesterday, so it can stand on its own without any preposition.

• Вчера мама сварила... = Yesterday Mom cooked...
• Сегодня мама сварила... = Today Mom cooked...

English also uses yesterday without a preposition, so this part matches English fairly well.

Yes. Russian word order is quite flexible.

Вчера мама сварила густой суп с фасолью и зелёным горохом is a neutral, natural sentence: it starts by setting the time, then gives the subject, then the action, then the object.

But other orders are possible, for example:

• Мама вчера сварила густой суп...
• Густой суп мама сварила вчера...

These versions may sound more marked or shift the emphasis, but the basic meaning stays the same because the case endings show the grammatical roles.

A helpful stressed version is:

вчера́ ма́ма свари́ла густо́й суп с фасо́лью и зелёным горо́хом

A few useful points:

• ё is pronounced like yo
• ё is always stressed
• so зелёным sounds roughly like zelyo-nym

Also:

• вчера́ = stress on the last syllable
• свари́ла = stress on и
• густо́й = stress on the last syllable
• фасо́лью = stress on со
• горо́хом = stress on ро

In many printed texts, ё is often written as е, but it is still pronounced yo when the word is actually зелёный.