На день рождения я хочу подарить маме тёплый шарф.

These two mean different things.

• на день рождения = for a birthday / as a birthday present / for the birthday occasion
• в день рождения = on the birthday itself (on that day)

So in this sentence, На день рождения tells you the occasion/purpose of the gift, not the exact day when the action happens.

Compare:

• На день рождения я купил торт. = I bought a cake for the birthday
• В день рождения я купил торт. = I bought a cake on the birthday

День рождения is a fixed expression meaning literally day of birth.

• день = day
• рождение = birth
• рождения = of birth (genitive)

So Russian expresses birthday as the day of birth.

In на день рождения, only день changes because it is the main noun after на:

• на день = for/on the day

But рождения stays genitive because it depends on день inside the phrase день рождения.

Because маме is in the dative case, which is used for the recipient of the gift.

With подарить (to give as a gift), Russian often uses this pattern:

• подарить кому? что?
• give to whom?
  • what?

So here:

• маме = to Mom
• тёплый шарф = a warm scarf

That is why it is:

• подарить маме шарф

not

• подарить маму шарф ❌

because маму would make Mom look like the direct object.

It is in the accusative, but for a masculine inanimate noun, the accusative singular looks the same as the nominative.

• nominative: шарф
• accusative: шарф

So тёплый шарф is the direct object, and it is accusative, but there is no visible change.

Compare with an animate masculine noun:

• Я вижу брата. = I see my brother

Here the accusative is different. But with шарф, nominative and accusative are identical.

This is about aspect.

• подарить = perfective
• дарить = imperfective

In this sentence, the speaker means a single completed action: giving one scarf as a present. That is why подарить is natural.

• Я хочу подарить маме шарф. = I want to give Mom a scarf (one complete act)

If you said хочу дарить, it would sound more like:

• wanting to give gifts repeatedly
• wanting to be in the process of giving
• a more habitual or ongoing idea

So for a one-time present, подарить is the usual choice.

Russian often leaves out possessive words like my, your, his, etc. when the meaning is obvious from context.

So:

• подарить маме usually means give Mom / give my mom
• English often needs my, but Russian often does not

If needed, Russian can say:

• моей маме = to my mom

But in many everyday sentences, that sounds unnecessary because the relationship is already clear.

Тёплый шарф is masculine singular, and the adjective agrees with the noun.

• шарф = masculine singular
• тёплый = masculine singular adjective form

In this sentence, the phrase is accusative singular, but since шарф is masculine inanimate, accusative looks like nominative:

• nominative: тёплый шарф
• accusative: тёплый шарф

So the adjective does not visibly change here.

ё shows both the sound and the stress.

• тёплый is pronounced roughly like TYOP-liy
• the stress falls on ё

In printed Russian, ё is sometimes replaced by е, so you may also see:

• теплый

But the correct pronunciation is still тёплый, not теплый with an unstressed e sound.

For learners, ё is very helpful because it tells you exactly where the stress is.

Yes, Russian word order is fairly flexible, because the cases show the grammatical roles.

The original sentence is neutral and natural:

• На день рождения я хочу подарить маме тёплый шарф.

But other orders are also possible, with different emphasis:

• Я хочу подарить маме тёплый шарф на день рождения.
  More like: I want to give Mom a warm scarf for her birthday.

• Маме я хочу подарить на день рождения тёплый шарф.
  Emphasizes Mom.

• Тёплый шарф я хочу подарить маме на день рождения.
  Emphasizes the warm scarf.

So the meaning stays similar, but the focus can shift.

Russian often leaves this unstated if the context makes it obvious.

So На день рождения can mean:

• for her birthday
• for my birthday
• for the birthday

In this sentence, since the gift is for маме, many people will naturally understand it as for Mom’s birthday, but the Russian wording itself does not force that interpretation without context.

If you want to be explicit, you can say:

• маме на день рождения = for Mom for her birthday / for Mom’s birthday
• на мамин день рождения = for Mom’s birthday

Yes, but the nuance is a bit different.

• на день рождения = for a birthday; very common for presents
• ко дню рождения = by the birthday / for the birthday occasion; can sound a bit more formal or goal-oriented

For example:

• Я купил маме шарф на день рождения. = I bought Mom a scarf for her birthday.
• Я подготовил подарок ко дню рождения. = I prepared the gift by the birthday / for the birthday occasion.

In your sentence, на день рождения is the most natural everyday choice.