На дереве было гнездо, и мама попросила нас говорить тише.

На дереве means on the tree, while в дереве would mean in the tree/inside the wood of the tree, which is not what is meant here.

For a bird’s nest, Russian normally says it is on a tree: гнездо на дереве.

Also, after на when talking about location, дерево changes to the Prepositional case:

• дерево → на дереве

Дереве is in the Prepositional case.

The basic noun is дерево. After на when it means in/on/at a place, Russian usually uses the Prepositional:

• на дереве = on the tree
• на столе = on the table
• в доме = in the house

So -е here is the normal Prepositional ending for this noun.

Было is the past tense of быть (to be) in the neuter singular form.

It agrees with гнездо, which is a neuter singular noun:

• гнездо = nest
• therefore: было гнездо

Compare:

• был дом — there was a house
• была книга — there was a book
• было гнездо — there was a nest

Russian often puts the location first in sentences like this when introducing something that exists somewhere.

So:

• На дереве было гнездо literally feels like On the tree there was a nest

This word order sounds natural when the speaker is presenting the scene.
If you said Гнездо было на дереве, that would also be grammatical, but it feels more like you are talking about the nest specifically and saying where it was.

Попросила is the past tense, feminine singular form of попросить (to ask/request).

It is feminine because the subject is мама, and мама is grammatically feminine:

• мама попросила = mom asked

Compare:

• папа попросил — dad asked
• мама попросила — mom asked

Because нас is the form used for the object of the verb.

In this sentence, мама is the one doing the asking, and us is the group being asked. So Russian uses the Accusative form:

• мы = we
• нас = us

The pattern is:

• попросить кого? — to ask whom?
• попросила нас — asked us

Because Russian commonly uses this pattern:

попросить + кого + infinitive

So:

• мама попросила нас говорить тише

This is a very normal and natural structure meaning mom asked us to speak more quietly.

Russian can also use чтобы in some kinds of subordinate clauses, but after попросить, the infinitive pattern is especially common and straightforward.

It is говорить because after попросила нас Russian uses an infinitive: to speak.

It is also imperfective, which fits the idea of speaking in a quieter way over some time, not just saying one thing once.

Compare:

• говорить тише = speak more quietly
• сказать тише would sound much less natural here, because сказать is more like say/utter once

So the sentence is about the general manner of speaking, not one single utterance.

Тише is the comparative form of тихо:

• тихо = quietly
• тише = more quietly / quieter

After verbs like говорить, Russian often uses the comparative when asking someone to lower the volume:

• Говори тише = Speak more quietly
• Можно тише? = Could it be quieter?

So говорить тише is the natural way to say speak more quietly.

Here it functions adverbially, because it describes how we should speak.

Even though the form comes from the comparative of тихий / тихо, in this sentence it modifies the verb говорить:

• говорить как? — тише
• speak how? — more quietly

So in English terms, it behaves like an adverb here.

Because и is joining two full clauses, each with its own grammatical base:

1. На дереве было гнездо
2. мама попросила нас говорить тише

In Russian, when и connects two independent clauses, a comma is normally used.

Yes. Russian word order is flexible, although different versions may shift the emphasis.

For example:

• На дереве было гнездо, и мама попросила нас говорить тише.
  This is natural and sets the scene first.

• Мама попросила нас говорить тише, потому что на дереве было гнездо.
  This makes the reason more explicit.

• Гнездо было на дереве, и мама попросила нас говорить тише.
  Grammatical, but the focus feels a bit different.

So the original sentence is not the only possible wording, but it is a very natural one.

Russian has no articles like a or the.

So гнездо can mean:

• a nest
• the nest

The exact meaning comes from context. In this sentence, English would usually translate it as a nest because it is being introduced for the first time.