Learn Hardcore German: Für ihre Entspannung macht meine Schwester abends Yoga, während ihr Freund leise Musik hört. - For her relaxation my sister does yoga in the evenings while her boyfriend listens to quiet music.

Entspannung is a feminine noun (die Entspannung).
After für, a feminine noun must be in the accusative: für die Entspannung.
With a possessive pronoun, that becomes für ihre Entspannung (her relaxation).

Ihrer Entspannung would be feminine dative or genitive, but für never uses those. So ihre Entspannung is correct here.

Who does ihre in für ihre Entspannung refer to?

In this sentence, ihre refers to meine Schwester.

So:

meine Schwester = my sister (subject)
für ihre Entspannung = for her relaxation (i.e. the sister’s relaxation)

It does not refer to ihr Freund here. Context and word order strongly suggest that the person relaxing is the sister.

If you wanted it to refer to the boyfriend, you would normally adjust the wording, for example:

Für seine Entspannung macht mein Schwager abends Yoga, während ihre Freundin leise Musik hört.
(Just as an example where possessives clearly line up with different people.)

What is the function of Für ihre Entspannung at the beginning of the sentence?

Für ihre Entspannung is a prepositional phrase of purpose: it tells us why she does yoga – for her relaxation.

Putting it at the very beginning:

Emphasizes the purpose.
Moves it into the Vorfeld (the first position) in German sentence structure.
Forces the finite verb (macht) into the second position:
Für ihre Entspannung (1) macht (2) meine Schwester abends Yoga (rest).

You could also place it later:

Meine Schwester macht abends Yoga für ihre Entspannung.
This is also correct, but the emphasis shifts slightly away from purpose and more onto the activity itself.

Why do we say macht ... Yoga instead of something like tut Yoga or spielt Yoga?

German typically uses machen with certain activities, including sports and similar practices:

Yoga machen – to do yoga
Sport machen – to do sports
Gymnastik machen – to do gymnastics

Using:

tun with Yoga sounds odd (Yoga tun is not idiomatic).
spielen is used for games, instruments, roles, etc. (Fußball spielen, Gitarre spielen), but not for Yoga.

So meine Schwester macht Yoga is the natural, idiomatic expression.

Why is there no article before Yoga? Why not ein Yoga or das Yoga?

With many sports and similar activities, German usually omits the article:

Ich mache Yoga.
Er spielt Fußball.
Sie macht Gymnastik.

Here, Yoga is treated more like an activity or practice, not a countable object. Saying ein Yoga would suggest one instance of a yoga thing, which sounds strange in normal usage.

An article is only used in special contexts, e.g.:

Das Yoga der 1970er Jahre (a specific type or historical phase of yoga)

What exactly does abends mean? How is it different from am Abend or jeden Abend?

abends is an adverb meaning in the evenings / at night (as a regular habit).

Comparisons:

abends
→ in the evening(s), generally, as a routine
Meine Schwester macht abends Yoga.
= She does yoga in the evenings (as a regular thing).
am Abend
→ on the evening (of a particular day) or in the evening in a more specific sense
Sie macht am Abend Yoga.
= She does yoga in the evening (today / on that specific evening).
jeden Abend
→ every evening, explicitly
Sie macht jeden Abend Yoga.
= She does yoga every single evening.

So abends suggests a habit without stressing “every single evening” as strongly as jeden Abend.

Why is the word order macht meine Schwester abends Yoga and not meine Schwester macht abends Yoga?

Actually, Meiner Schwester macht abends Yoga would be wrong (wrong case on the noun), but:

Meine Schwester macht abends Yoga.
Abends macht meine Schwester Yoga.
Für ihre Entspannung macht meine Schwester abends Yoga.

are all correct.

German main clauses require the finite verb in second position:

If meine Schwester is first: Meine Schwester (1) macht (2) …
If für ihre Entspannung is first: Für ihre Entspannung (1) macht (2) …

In your sentence, Für ihre Entspannung is in first position, so the subject meine Schwester comes after the verb:
Für ihre Entspannung macht meine Schwester abends Yoga.

What does während do in this sentence, and why does the verb hört go to the end?

während is a subordinating conjunction meaning while. It introduces a subordinate clause that happens at the same time as the main clause.

Main clause: Für ihre Entspannung macht meine Schwester abends Yoga
Subordinate clause: während ihr Freund leise Musik hört

In German, subordinating conjunctions like während, weil, dass, wenn send the finite verb to the end of the clause:

… während ihr Freund leise Musik hört.
(verb hört at the end)

So the pattern is:

während
- subject + (objects/adverbs) + verb at the end

Why is it ihr Freund and not ihrer Freund or something with an article like der Freund?

ihr Freund here means her boyfriend / her male partner / her (male) friend.

ihr is a possessive pronoun (her)
Freund is masculine, and in this clause it is the subject (nominative):
- wer? / wer hört? → ihr Freund

In the nominative masculine singular, the possessive pronoun is simply:

ihr Freund (her friend)
sein Freund (his friend)
mein Freund (my friend)

ihrer Freund would be a different case (genitive or dative plural context), so it does not fit here.

Also, you do not need an extra article:

ihr Freund already expresses the friend belonging to her, so der would be redundant (der ihr Freund is wrong in standard German).

Could ihr Freund also mean your friend (formal “you”)?

Yes, in writing Ihr Freund (with capital I) can mean your friend (addressing someone politely with Sie):

Ihr Freund (capital I) = your friend (formal you)
ihr Freund (lowercase i) = her friend

In your sentence, ihr is lowercase, so it clearly means her. In normal texts, context and capitalization tell you which meaning is intended.

Why is it leise Musik and not leiser Musik or eine leise Musik?

Here, leise Musik is:

singular, feminine, accusative (Musik is a feminine noun)
without an article
with an attributive adjective leise

In this pattern (feminine, accusative, no article), the adjective ending is -e:

leise Musik (correct: strong declension)
leiser Musik would be dative or genitive, which is not the case here.

About the article:

Musik is often used as a mass noun (not usually counted).
Saying leise Musik = soft music (in general), which is very natural.
eine leise Musik is possible but more stylistic or poetic, often referring to a specific kind or instance of music as an almost countable thing. In everyday language, leise Musik is more common.

Why do we say leise Musik hört and not hört leise Musik in the während-clause?

Because während introduces a subordinate clause, where the finite verb goes to the end:

Main clause word order: Verb in 2nd position
- Ihr Freund hört leise Musik.
Subordinate clause with während: Verb at the end
- … während ihr Freund leise Musik hört.

So:

hört leise Musik would be correct word order only in a main clause
In the während-clause, the verb must move to the final position: leise Musik hört

Could I say Um sich zu entspannen, macht meine Schwester abends Yoga instead of Für ihre Entspannung macht …? Is there a difference?

Yes, you can say:

Um sich zu entspannen, macht meine Schwester abends Yoga.
= In order to relax, my sister does yoga in the evening.

Differences in nuance:

Für ihre Entspannung
- Uses a noun (Entspannung) and für
  - accusative.
- Literally: for her relaxation → a bit more noun-based and slightly more formal-sounding.
Um sich zu entspannen
- Uses a purpose construction with um … zu + infinitive.
- Literally: in order to relax → more verb-based, very common and natural in speech and writing.

Both are grammatically correct and natural; the um … zu version sounds a bit more direct and typical for talking about purpose in everyday language.