Повесь жилетку на вешалку, пожалуйста: в комнате уже тепло.

Повесь is the correct imperative form of повесить (to hang up / to put up).

Many Russian imperatives are not formed in a fully predictable way from the infinitive, so you usually need to learn them as forms:

• повесить → повесь (singular informal command)
• повесить → повесьте (plural or polite command)

So:

• Повесь жилетку = Hang up the vest
• Повесьте жилетку = Hang up the vest (to more than one person, or politely to one person)

This is a very common aspect question.

• повесь comes from the perfective verb повесить
• вешай comes from the imperfective verb вешать

In commands:

• perfective often means do it once / bring the action to completion
• imperfective often means be doing it / do it as a process / do it regularly

So here:

• Повесь жилетку = Hang up the vest (one completed action)
• Вешай жилетку would sound more like Go ahead and hang the vest up or could fit a context where the process is being emphasized.

In this sentence, the speaker wants a single completed action, so повесь is the natural choice.

Because жилетку is the accusative case, used for the direct object of the verb.

The dictionary form is:

• жилетка = vest, waistcoat

But after a transitive verb like повесить (to hang up something), the thing being hung is in the accusative:

• повесить жилетку

Since жилетка is an inanimate feminine noun ending in -а, its accusative singular changes to -у:

• nominative: жилетка
• accusative: жилетку

Because Russian uses different cases depending on whether there is motion toward a place or location in a place.

Here, на вешалку means onto the hanger / onto the coat rack, so it shows direction/result of movement, and that requires the accusative.

Compare:

• Повесь жилетку на вешалку. = Hang the vest on the hanger.
  (movement onto it)
• Жилетка висит на вешалке. = The vest is hanging on the hanger.
  (location, so prepositional case)

So:

• на + accusative = motion onto
• на + prepositional = location on

Вешалку is accusative singular of вешалка.

Like жилетка, it is a feminine noun ending in -а, so the accusative singular changes to -у:

• nominative: вешалка
• accusative: вешалку

It is accusative here because it follows на in the sense of motion toward a destination:

• на вешалку = onto the hanger / coat rack

Вешалка can mean several related things depending on context:

• a hanger
• a coat rack
• a place for hanging clothes

In this sentence, it most likely means hanger or coat rack, depending on the situation. Russian often leaves this a bit broader than English.

If the speaker means the kind of hanger you put clothes on in a wardrobe, Russians also often say плечики.

The colon introduces an explanation or reason for the command.

So the sentence structure is basically:

• Hang up the vest, please: it’s already warm in the room.

In English, you might also express this with:

• ..., because it’s already warm in the room.

Russian punctuation often uses a colon in places where English might prefer because, a dash, or a new sentence.

Yes. You could say:

• Повесь жилетку на вешалку, пожалуйста, потому что в комнате уже тепло.

That is completely understandable and natural.

The version with the colon:

• Повесь жилетку на вешалку, пожалуйста: в комнате уже тепло.

sounds a bit more compact and written-style, with the second part clearly presented as the reason.

This is normal in Russian.

Russian often omits the present-tense form of to be. So where English says:

• It is warm in the room

Russian simply says:

• В комнате тепло

There is no present-tense есть here.

So:

• в комнате уже тепло literally looks like in the room already warm
• but it naturally means it’s already warm in the room

In this sentence, тепло is not an ordinary adjective modifying a noun. It is a predicative word or word of state.

It expresses a condition:

• Здесь тепло. = It’s warm here.
• Мне холодно. = I’m cold.
• В комнате тихо. = It’s quiet in the room.

So тепло here means it is warm, not warm attached to a noun.

Уже usually means already, but in context it can also feel like now already / by now.

In this sentence:

• в комнате уже тепло = the room is already warm

The idea is that the room has reached a warm enough temperature, so the extra clothing is no longer needed.

Because в means in, and комнате is the prepositional singular of комната (room).

After в when talking about location, Russian uses the prepositional case:

• в комнате = in the room

Forms:

• nominative: комната
• prepositional: в комнате

Пожалуйста makes the command more polite.

Compare:

• Повесь жилетку на вешалку. = a plain command
• Повесь жилетку на вешалку, пожалуйста. = Please hang the vest on the hanger.

It softens the imperative and makes it sound more courteous.

Because пожалуйста is treated like a parenthetical politeness word in this kind of sentence.

So it is commonly set off with commas:

• Повесь жилетку на вешалку, пожалуйста.
• Пожалуйста, повесь жилетку на вешалку.
• Повесь, пожалуйста, жилетку на вешалку.

All of these are possible. The commas show that пожалуйста is not grammatically part of the main clause in the same way as the verb or object.

Yes. Russian word order is flexible, though the emphasis changes.

Possible variants:

• Повесь жилетку на вешалку, пожалуйста.
• Повесь, пожалуйста, жилетку на вешалку.
• Жилетку повесь на вешалку, пожалуйста.

These all mean roughly the same thing, but they highlight different parts:

• starting with повесь focuses on the action
• starting with жилетку gives more emphasis to the vest itself

The original order is very natural and neutral.

It is a normal word, but the exact English equivalent depends on context.

Жилетка can mean:

• vest
• waistcoat
• sometimes a sleeveless upper garment more generally

Russian clothing words do not always match English clothing categories one-to-one, so the best translation depends on what kind of garment is meant in context.

The main stresses are:

• повЕсь
• жилЕтку
• на вЕшалку
• в кОмнате
• ужЕ
• теплО

Stress is important in Russian, and it is not always predictable, so it is worth learning words together with their stress:

• жилЕтка
• вЕшалка
• кОмната
• теплО

Yes, if you are speaking:

• to more than one person, or
• to one person politely

Compare:

• Повесь жилетку... = singular informal
• Повесьте жилетку... = plural or polite singular

So the sentence as given sounds like it is addressed informally to one person, for example a friend, family member, or child.