На большом плакате дети написали свои имена, а под ним повесили гирлянду.

Because на here means on, expressing location, so Russian uses the prepositional case.

• плакат → dictionary form
• на плакате = on the poster

The adjective also has to match the noun:

• большой плакат = a big poster
• на большом плакате = on a big poster

So both words change because they are in the prepositional case after на when talking about location.

It changes to agree with плакате in:

• gender: masculine
• number: singular
• case: prepositional

The full phrase is:

• nominative: большой плакат
• prepositional: на большом плакате

This is standard adjective agreement in Russian.

Russian often uses the reflexive possessive свой when the thing belongs to the subject of the clause.

Here, the subject is дети (the children), and they wrote their own names, so Russian prefers:

• дети написали свои имена

Using их имена could sound like the names of some other people, depending on context.

So:

• свои имена = their own names
• их имена = their names, possibly someone else’s

Because it agrees with имена, which is plural.

• singular: своё имя = one’s own name
• plural: свои имена = one’s own names

Even though свой refers back to дети, its form is determined by the noun it describes, not directly by the owner.

Because имена is the accusative plural of имя, and here it is the direct object of написали.

• имя = name
• имена = names

In this case, for this neuter plural noun, the accusative looks like the nominative plural.

So:

• дети написали свои имена = the children wrote their names

Because the subject is plural: дети (children).

In the Russian past tense, verbs agree with the subject in gender/number:

• он написал = he wrote
• она написала = she wrote
• они написали = they wrote

And similarly:

• они повесили = they hung

So both verbs are plural because the action is done by more than one person.

They describe completed actions:

• the children wrote the names
• then they hung a garland

The perfective aspect is natural here because the sentence is about finished results.

Compare:

• писали = were writing / wrote (process, repeated, or ongoing)

• написали = wrote, finished writing

• вешали = were hanging / used to hang
• повесили = hung, completed the action

So the sentence presents two completed events.

Ним means it/him in a case form used after certain prepositions. Here it refers to плакат (poster).

• под ним = under it

Because плакат is masculine, the pronoun is based on он:

• nominative: он
• instrumental: им
• after a preposition: ним

So:

• под ним literally = under it

Because под can take different cases depending on meaning.

For location (under, stationary position), Russian usually uses the instrumental case:

• под столом = under the table
• под ним = under it

If it meant movement to a position under something, Russian could use the accusative in some contexts. But here the garland ends up located under the poster, so под ним with instrumental is the normal choice.

The subject is understood from the first clause: дети.

Russian often leaves out the subject in the second part of a sentence when it is the same and already obvious:

• Дети написали свои имена, а под ним повесили гирлянду.

This is similar to English:

• The children wrote their names on the poster and hung a garland under it.

Russian simply does not need to repeat дети.

А often links two parts while also marking a slight contrast, shift, or change of focus.

Here the sentence moves from one action to another:

• on the poster, they wrote their names,
• and under it, they hung a garland.

Using а helps separate the two coordinated ideas a bit more clearly than и would.

Very roughly:

• и = and
• а = and / while / whereas / and then, with a contrast or shift

In this sentence, а sounds natural because the second action is in a different place: под ним instead of на ... плакате.

Because it is the direct object of повесили.

• гирлянда = garland
• повесили гирлянду = hung a garland

For feminine nouns ending in -а, the accusative singular usually changes to -у:

• гирлянда → гирлянду

Yes. Russian word order is flexible, although different orders sound more or less natural depending on emphasis.

The given version highlights the locations nicely:

• На большом плакате дети написали свои имена, а под ним повесили гирлянду.

Other possible versions include:

• Дети написали свои имена на большом плакате, а под ним повесили гирлянду.
• Дети на большом плакате написали свои имена, а под ним повесили гирлянду.

These all mean roughly the same thing, but the original puts emphasis on where the actions happened.

Russian has no articles like English a or the.

So:

• на большом плакате can mean on a big poster or on the big poster
• повесили гирлянду can mean hung a garland or sometimes hung the garland, depending on context

Russian relies on context, word order, and shared knowledge instead of articles.

Because the two locations are different:

• на плакате = on the poster
• под ним = under it

Russian chooses the preposition based on the spatial relationship:

• на = on, on the surface of
• под = under, beneath

So the sentence describes two different places relative to the same poster: the names are on it, and the garland is under it.