Learn Hardcore Finnish: Ystäväni pitää eniten romanttisista elokuvista, erityisesti kun kyse on kevyestä komediasta. - My friend likes romantic movies the most, especially when it is a light comedy.

Only context will tell which is intended. With this sentence alone, my friend is the more natural default interpretation, but my friends is not grammatically impossible.

Why is it pitää here? I thought pitää means to hold.

Finnish pitää is a very polysemous verb. Two common meanings are:

pitää + object (accusative) → to hold, to keep
- Pidän tätä kirjaa kädessäni. = I am holding this book in my hand.
pitää jostakin (elative case -sta/-stä) → to like something
- Pidän suklaasta. = I like chocolate.

In your sentence:

pitää romanttisista elokuvista = to like romantic movies

So here pitää is the "to like" verb, and that usage always requires the thing liked to be in the elative case (ending in -sta / -stä).

Why is it romanttisista elokuvista and not something like romanttiset elokuvat?

The form comes from two separate requirements:

The verb pitää (to like) requires jostakin, meaning the noun must be in the elative case.
- Base: romanttinen elokuva (a romantic movie)
- Plural stem: romanttiset elokuvat (romantic movies, nominative plural)
- Elative plural: romanttisista elokuvista (from/of romantic movies)
Because of the elative case, both words change:
- romanttinen → romanttisista (adjective in elative plural)
- elokuva → elokuvista (noun in elative plural)

So romanttisista elokuvista is “from romantic movies” literally, but with pitää jostakin it means “(likes) romantic movies”.

What does eniten mean, and where does it come from?

Eniten means the most.

It is the superlative form in the comparative series:

paljon = much / a lot
enemmän = more
eniten = the most

In this sentence:

pitää eniten romanttisista elokuvista
= (he/she) likes romantic movies the most

So eniten modifies how much the friend likes something, indicating that romantic movies are their top preference.

Could the word order be Ystäväni pitää romanttisista elokuvista eniten? Would that change the meaning?

Yes, you can say:

Ystäväni pitää romanttisista elokuvista eniten.

This is perfectly grammatical and the meaning is essentially the same: My friend likes romantic movies the most.

Differences are subtle and mostly about emphasis and rhythm:

pitää eniten romanttisista elokuvista
→ slightly emphasizes how much (the "most") first.
pitää romanttisista elokuvista eniten
→ presents the thing liked (romantic movies) earlier, then adds that it is the most.

Both are very natural in everyday Finnish.

Why is erityisesti used here? What nuance does it add?

Erityisesti means especially, particularly.

In this sentence:

Ystäväni pitää eniten romanttisista elokuvista, erityisesti kun kyse on kevyestä komediasta.

The first part already says your friend likes romantic movies the most. Erityisesti adds a further nuance:

They like romantic movies the most in general,
but especially when those romantic movies are light comedies.

So erityisesti narrows down and highlights a specific subcategory of romantic movies as an extra strong preference.

What does the expression kun kyse on mean?

Kun kyse on is a very common fixed expression meaning:

when it is a question of
when it is about
when the issue is

Breaking it down:

kun = when
kyse = issue, matter, what is at stake
on = is (3rd person singular of olla, to be)

So:

erityisesti kun kyse on kevyestä komediasta
≈ especially when it is about light comedy
≈ especially when it comes to light comedy

Why is kevyestä komediasta also in the -sta/-stä form? What case is that?

Kevyestä komediasta is again in the elative case (ending in -sta / -stä):

Base phrase: kevyt komedia = light comedy
Elative singular: kevyestä komediasta = from light comedy / about light comedy

This form is required by the expression kyse on jostakin:

kyse on jostakin = it is about something
The jostakin part must be in elative.

So:

kun kyse on kevyestä komediasta
= when it is about light comedy,
where kevyestä komediasta is the thing the matter is about, marked in elative.

What is the difference between kevyt and kevyestä?

They are forms of the same adjective kevyt (light).

kevyt = base form (nominative singular)
- kevyt komedia = light comedy (as a basic noun phrase)
kevyestä = elative singular of kevyt
- Used because the whole phrase kevyestä komediasta is in the elative case.
- Adjectives agree in case and number with the noun they modify.

So when komedia becomes komediasta (elative), kevyt must also change to kevyestä to match that case.

What is the difference between pitää jostakin and tykätä jostakin? Could I say Ystäväni tykkää eniten romanttisista elokuvista?

Both verbs are very common and very similar in meaning:

pitää jostakin = to like something
tykätä jostakin = to like something

Differences:

pitää is slightly more neutral and is used in both spoken and written language.
tykätä is also very common, feels just a bit more colloquial or casual, especially in some regions.

Your alternative:

Ystäväni tykkää eniten romanttisista elokuvista, erityisesti kun kyse on kevyestä komediasta.

is perfectly fine and natural Finnish. The grammar with elative (-sta/-stä) remains the same: tykätä jostakin also requires elative.

Why is there no personal pronoun like hän in the sentence? How do we know who pitää refers to?

In Finnish, the personal pronoun is often dropped when the subject is clear from context or morphology.

Here:

The subject is Ystäväni (my friend).
The verb pitää is in 3rd person singular, which fits with ystäväni.

So:

Ystäväni pitää ... = My friend likes ...

No need to add hän (he/she):

Ystäväni hän pitää ... would be redundant and sounds unnatural in standard Finnish (though you might hear similar structures in some dialects with a different word order and nuance).

Could the sentence be shortened while keeping the same meaning? What would be a simpler version?

A simpler but still natural version could be:

Ystäväni pitää eniten romanttisista elokuvista, erityisesti kevyestä komediasta.

Here we:

Drop kun kyse on, and instead directly say:
- erityisesti kevyestä komediasta = especially (romantic movies that are) light comedies

The nuance is almost the same: it still tells us that among romantic movies, those that are light comedies are your friend's particular favorite.