Learn Hardcore Croatian: Ovaj san mi je važan. - This dream is important to me.

Without mi, Ovaj san je važan just means This dream is important (in general).
With mi, you add a personal nuance: it matters specifically to me.

Why is it mi and not meni?

Both mi and meni mean to me, but:

mi = clitic (short, unstressed form), normally placed in the “clitic slot” in the sentence (Ovaj san *mi je važan*).
meni = stressed form (used when you want to emphasize me specifically).

Compare:

Ovaj san mi je važan. – This dream is important to me. (neutral)
Ovaj san je meni važan. – This dream is important to me (as opposed to others).

So, mi is the normal, unstressed choice here; meni is for emphasis or contrast.

Why is it ovaj and not ovo or ova?

Croatian demonstratives must agree with the noun in gender, number, and case.

san (dream) is masculine, singular, nominative.
Masculine singular nominative form of ovaj/ova/ovo is ovaj.

Forms:

ovaj – masculine (e.g. ovaj san, ovaj čovjek)
ova – feminine (e.g. ova knjiga, ova žena)
ovo – neuter (e.g. ovo dijete, ovo pismo)

So you must say ovaj san, not ovo san or ova san.

What case is san in, and why?

San is in the nominative singular.

Reason: San is the subject of the sentence:

Ovaj san (subject) mi je važan (predicate).

Nominative is used for the subject (who/what is being talked about).
Since we’re just stating a characteristic of the dream (that it’s important), the subject stays in nominative.

Why does važan end in -an? What is it agreeing with?

Važan is an adjective meaning important. In Croatian, adjectives must agree with the noun they describe in gender, number, and case.

san = masculine, singular, nominative
Therefore the adjective must be masculine singular nominative: važan.

Other forms of the same adjective:

važna knjiga – important book (feminine, singular, nominative)
važno dijete – important child (neuter, singular, nominative)

In our sentence, važan agrees with san:
Ovaj san mi je važan. (masc. sg. nom. + masc. sg. nom.)

Can I say Ovaj san je mi važan or Ovaj san važan mi je?

No. Croatian clitics like mi have very strict word order rules.

Correct:

Ovaj san mi je važan.
Ovaj mi je san važan.
Meni je ovaj san važan. (using meni, not mi, and putting it in a stressed position)

Incorrect / very unnatural:

Ovaj san je mi važan. ✗
Ovaj san važan mi je. ✗

Normally, when you have a subject at the beginning, clitics (like mi, je) appear early in the sentence in a fixed slot, as in Ovaj san mi je važan.

What is the difference between Ovaj san mi je važan and Meni je ovaj san važan?

Both mean essentially the same: This dream is important to me.

The differences are subtle and about emphasis:

Ovaj san mi je važan. – Neutral word order, light “to me” nuance.
Meni je ovaj san važan. – Emphasizes me: To me, this dream is important (maybe in contrast to what others think).

So mi is neutral, meni moves to the front when you want to highlight/contrast the person.

Could I simply say Ovaj san je važan za mene instead of using mi?

Yes, but the nuance changes slightly.

Ovaj san mi je važan. – Natural, everyday way to say This dream is important to me, using the dative mi.
Ovaj san je važan za mene. – Literally This dream is important for me. This can sound a bit more explicit or heavy, sometimes with a sense of benefit or significance for my situation.

Both are correct, but dative + mi is more idiomatic and common for “important to me”.

How is san related to sanjati or sanjam?

san – noun: a dream
sanjati – infinitive verb: to dream
sanjam – 1st person singular present of sanjati: I dream

Examples:

Sanjam ovaj san svake noći. – I dream this dream every night.
Ovaj san mi je važan. – This dream is important to me.

So in our sentence, san is just the noun, not the verb form.

Can the word order change to Važan mi je ovaj san? Does it mean the same?

Yes, Važan mi je ovaj san is correct and means the same basic thing: This dream is important to me.

Word order in Croatian is flexible and used for emphasis:

Ovaj san mi je važan. – Neutral; topic is “this dream.”
Važan mi je ovaj san. – Emphasizes važan (important): Important to me is this dream (in particular).

All of these are grammatical, but they highlight different parts of the sentence.

How would the sentence change if the noun were feminine or neuter?

The adjective and demonstrative would change to match the noun’s gender.

Feminine example (knjiga, book):

Ova knjiga mi je važna. – This book is important to me.
- ova (fem.)
- važna (fem.)

Neuter example (pismo, letter):

Ovo pismo mi je važno. – This letter is important to me.
- ovo (neut.)
- važno (neut.)

With san (masculine), we correctly have:

Ovaj san mi je važan.