Ta grana dodiruje prozor kad puše jak vjetar.

Ta means that. It is a demonstrative word, and it has to agree with grana in:

• gender: feminine
• number: singular
• case: nominative

So ta grana means that branch.

It is in the nominative because grana is the subject of the sentence. If it were an object, you would see a different form, for example tu granu.

Because prozor is a masculine inanimate noun, and in Croatian the accusative singular of masculine inanimate nouns is the same as the nominative singular.

So:

• nominative: prozor
• accusative: prozor

That is why the sentence says dodiruje prozor.

This is different from masculine animate nouns, where the accusative usually changes, for example:

• nominative: čovjek
• accusative: čovjeka

Dodiruje is the 3rd person singular present tense form, meaning touches or is touching.

Here it matches the subject ta grana:

• ja dodirujem = I touch
• ti dodiruješ = you touch
• on/ona/ono dodiruje = he/she/it touches

In this sentence, it is used for a general repeated situation: whenever the strong wind blows, the branch touches the window.

This is about aspect, which is very important in Croatian.

• dodiruje is imperfective: it describes an ongoing, repeated, or habitual action
• dotakne is perfective: it describes a single completed touch

In this sentence, the idea is a regular situation, not one single event. So dodiruje fits better.

Very roughly:

• Ta grana dodiruje prozor kad puše jak vjetar.
  = This happens in general / repeatedly.

• Ta grana dotakne prozor kad zapuše jak vjetar.
  = The branch touches the window when a strong wind starts blowing / on a particular occasion.

Yes. Both mean when.

The difference is mostly style:

• kad = shorter, very common in everyday speech and writing
• kada = a bit fuller, sometimes slightly more formal or emphatic

So you could also say:

Ta grana dodiruje prozor kada puše jak vjetar.

Both are correct.

Here puše comes from puhati, which means to blow.

So:

• puhati = to blow
• puše = blows / is blowing

A useful warning: puše can look confusing because there is also pušiti, which means to smoke. But in this sentence it clearly means blows, because the subject is jak vjetar.

Because Croatian word order is much more flexible than English word order.

Both of these are possible:

• kad puše jak vjetar
• kad jak vjetar puše

The version with the verb first sounds very natural here.

Also, unlike English, Croatian does not need a dummy subject like it. English says when a strong wind blows, but Croatian can simply say kad puše jak vjetar, with the real subject jak vjetar coming after the verb.

In masculine singular nominative, Croatian adjectives can have a short form and a long form.

Here:

• jak = short form
• jaki = long form

In everyday neutral description, jak vjetar is the most natural choice.
Jaki vjetar is also possible, but it can sound a bit more definite, emphatic, or stylistically different.

For a learner, the safest basic phrase here is:

jak vjetar = strong wind

Because in a sentence like this, when the kad clause comes after the main clause, Croatian normally does not need a comma there.

So this is natural:

Ta grana dodiruje prozor kad puše jak vjetar.

But if the kad clause comes first, then you do use a comma after it:

Kad puše jak vjetar, ta grana dodiruje prozor.

Because Croatian has no articles.

English uses a and the, but Croatian usually does not. Whether something is definite or indefinite is understood from context, word order, or words like demonstratives.

So:

• prozor can mean a window or the window, depending on context
• vjetar can mean wind, a wind, or the wind, depending on context

Notice that ta is not an article. It specifically means that. So ta grana is not just the branch, but more specifically that branch.