После дождя на дороге была большая лужа, а вокруг была грязь.

Because после always requires the genitive case.

• дождь = nominative singular, the dictionary form
• после дождя = after the rain

This is a very common pattern:

• после урока = after the lesson
• после работы = after work
• после обеда = after lunch

So in this sentence, после дождя is the normal and required form.

Here на дороге means on the road or in the road, describing location, so Russian uses the prepositional case after на when talking about where something is.

• дорога = road
• на дороге = on the road

Compare:

• на дороге была лужа = there was a puddle on the road
• на дорогу = onto the road, toward the road

So:

• где? where? → на дороге
• куда? to where? → на дорогу

Была is the past tense of быть and agrees with the noun it goes with.

In Russian past tense, the verb agrees in gender and number:

• был = masculine
• была = feminine
• было = neuter
• были = plural

Here:

• лужа is feminine → была большая лужа
• грязь is also feminine → была грязь

So both parts use была.

It is repeated because the sentence has two separate ideas joined by а:

• there was a big puddle on the road
• and around there was mud

Russian can sometimes omit repeated words, but here repeating была sounds natural and clear.

Because adjectives must agree with the noun in gender, number, and case.

• лужа is feminine singular nominative
• so большой must become большая

Examples:

• большой дом = a big house (masculine)
• большая лужа = a big puddle (feminine)
• большое окно = a big window (neuter)
• большие лужи = big puddles (plural)

Here вокруг means around, all around, or in the surrounding area.

In this sentence:

• а вокруг была грязь = and there was mud all around

It does not need another noun here because it works adverbially: it describes the surrounding space in a general way.

You can also see вокруг used with a noun in the genitive:

• вокруг дома = around the house
• вокруг озера = around the lake

But in your sentence, it stands alone and means something like all around.

Both can be translated as and, but they are not always interchangeable.

• и simply adds one fact to another
• а often introduces a contrast, shift, or separate parallel detail

In this sentence, а sounds natural because the speaker is moving from one image to another:

• on the road there was a big puddle,
• and meanwhile / while / and around it there was mud.

So а helps separate the two parts a little more than и would.

Грязь usually works as an uncountable mass noun, like mud or dirt in English.

So была грязь means:

• there was mud
• it was muddy
• there was dirt all around

The form грязи would not fit here as the basic subject of the sentence. You use грязи in other grammatical contexts, for example:

• много грязи = a lot of mud
• без грязи = without mud

But in your sentence, грязь is the subject, so nominative грязь is correct.

Russian has no articles like English a/an/the.

So:

• была большая лужа can mean there was a big puddle
• depending on context, it could also feel like there was the big puddle, but usually English would say a big puddle

Russian leaves definiteness to context, word order, and common sense.

That is why the sentence has no separate word for a before большая лужа or mud before грязь.

Russian word order is more flexible than English, although different orders can sound more natural in different contexts.

The given sentence is very natural:

• После дождя на дороге была большая лужа, а вокруг была грязь.

But other orders are possible, for example:

• После дождя большая лужа была на дороге, а вокруг была грязь.
• На дороге после дождя была большая лужа, а вокруг была грязь.

These are grammatically possible, but the original order sounds smoother and more neutral.

In Russian, word order often helps with focus:

• earlier position = background/topic
• later position = newer or more important information

So на дороге была большая лужа naturally presents a big puddle as the new information.

Sometimes Russian can omit repeated words, but here keeping была is the most natural choice.

• ...а вокруг была грязь sounds complete and balanced.
• ...а вокруг грязь is possible in some contexts, but it sounds more like a present-tense observation or a more abrupt style.

Since the first clause is in the past, repeating была keeps the tense and structure very clear.

Russian often puts a time phrase near the beginning to set the scene.

• После дождя = after the rain

This tells you the situation first, before describing what was there.

This is very natural Russian style, similar to English sentences like:

• After the rain, there was a big puddle on the road...

You could move it, but the beginning is the most natural place for this kind of background information.

A natural reading would usually stress the main new information:

После дождя на дороге была большАя лУжа, а вокрУг была грЯзь.

The strongest stress often falls on:

• лужа
• грязь

because those are the key things being described.

If the speaker especially wants to emphasize size, then большая may get extra stress too.

Yes. In each clause, the noun is the grammatical subject:

• на дороге была большая лужа
• вокруг была грязь

This is a very common Russian pattern for introducing what existed somewhere in the past:

• В комнате был стол. = There was a table in the room.
• На улице были люди. = There were people outside.
• На дороге была большая лужа. = There was a big puddle on the road.

So even though English uses there was, Russian simply uses the noun as the subject with the verb.

Yes, depending on context.

Грязь can mean:

• mud
• dirt
• filth
• mess

In this sentence, because it begins with После дождя (after the rain), mud is usually the best translation. Rain makes the road muddy, so mud sounds more natural than dirt here.