Learn Hardcore Japanese: huyu ha atatakai bousi wo kaburimasu.

If you really want to say I, you can add it:
私は冬は暖かい帽子をかぶります。 – As for me, in winter I wear a warm hat.
But in everyday Japanese, the shorter version without 私 is more natural if context makes it clear.

What does the particle は after 冬 do? Is it the same as “in” in English?

は marks the topic of the sentence: what you’re talking about.
In 冬は暖かい帽子をかぶります, it literally means “As for winter…”.

English doesn’t have a direct equivalent, so we translate it as “In winter” because that sounds natural.
The nuance is: speaking about winter (as opposed to other seasons), I wear a warm hat.
So 冬は is not “in winter” grammatically, but it often gets translated that way.

What’s the difference between 冬は, 冬に, and 冬には?

冬は暖かい帽子をかぶります。
“In winter, I wear a warm hat.”
Winter is the topic. Often used for general, habitual statements and contrasts like:
冬は暖かい帽子をかぶりますが、夏はTシャツを着ます。
冬に暖かい帽子をかぶります。
“I wear a warm hat in winter.”
に marks time (in winter / during winter) more neutrally. Used for events or habits without strong contrast.
冬には暖かい帽子をかぶります。
“(Especially) in winter, I wear a warm hat.”
には = に (time) + は (contrast/topic). It emphasizes in winter (as opposed to other times).

All three can be correct; the differences are nuance and emphasis.

Why does 帽子 take を? I thought を was just for direct objects.

It is for direct objects, and here 帽子 is the direct object of かぶります.
The verb かぶる means “to put on / wear (on the head)”, so you are doing something to the hat.

Think of it like English “put on a hat / wear a hat” — a hat is also the object in English.
So:

帽子をかぶります。 – I wear/put on a hat.
コートを着ます。 – I wear/put on a coat.

In both cases, the clothing item takes を because it’s what the action is done to.

What exactly does かぶります mean? Is it just “wear”?

かぶります is the polite form of かぶる. It means:

To put on or wear something on your head (hat, cap, helmet, etc.)

Japanese has different verbs for “wear” depending on the body part:

かぶる – hats, caps, helmets (on the head)
- 帽子をかぶります。 – I wear a hat.
着る（きる） – shirts, jackets, dresses, coats (upper body / full body)
- コートを着ます。 – I wear a coat.
はく – pants, skirts, shoes, socks (lower body / feet)
- 靴をはきます。 – I wear shoes.
する – accessories like earrings, necklaces, neckties, watches
- ネクタイをします。 – I wear a tie.
かける – glasses
- 眼鏡をかけます。 – I wear glasses.

So かぶる is not a general “wear” verb; it’s specifically for headwear.

Why is 暖かい used here instead of あつい?

Both relate to heat, but the nuance is different:

暖かい（あたたかい） – warm in a pleasant, comfortable way.
Used for weather, rooms, clothing, people’s hearts, etc.
- 暖かい帽子 – a warm (cozy) hat
- 今日は暖かいですね。 – It’s warm today, isn’t it?
熱い（あつい） – hot, usually physically hot or too hot.
Often something you don’t want to touch or that feels intense.
- 熱いお茶 – hot tea
- この鍋は熱いです。 – This pot is hot.

A hat that keeps you comfortably warm is 暖かい, not 熱い or あつい.

Why doesn’t 暖かい have a particle after it?

Here 暖かい is an adjective directly modifying the noun 帽子:

暖かい帽子 – a warm hat (just like English “warm hat”)

In Japanese, adjectives go before the noun they modify, and you don’t need a particle between them.

Compare:

暖かい帽子をかぶります。 – I wear a warm hat. (adjective + noun)
帽子は暖かいです。 – The hat is warm. (noun + は + adjective + です)

When the adjective describes a noun directly (warm hat), no particle.
When it’s in a “X is Y” type sentence (the hat is warm), you use は or が after the noun.

Is this sentence polite or casual? How would it look in casual or past tense?

かぶります is the polite non-past form, so the whole sentence is polite:

冬は暖かい帽子をかぶります。 – polite

Some common variations:

Casual present/habitual:
冬は暖かい帽子をかぶる。
Polite past:
冬は暖かい帽子をかぶりました。 – I wore a warm hat (in winter).
Casual past:
冬は暖かい帽子をかぶった。
Polite negative:
冬は暖かい帽子をかぶりません。 – I don’t wear a warm hat in winter.

The –ます ending is the main marker of politeness here.

Does 冬は暖かい帽子をかぶります mean I do it once, or regularly?

The non‑past form (かぶります) in Japanese covers:

present
future
habitual actions

In this sentence, because the time expression is a season (冬), the most natural reading is habitual:

“In winter, I (usually / generally) wear a warm hat.”

If you wanted to talk about what you’re doing right now, you’d normally use the progressive:

今、暖かい帽子をかぶっています。 – I am wearing a warm hat now.

Can I change the word order, like 暖かい帽子は冬にかぶります? Is that still correct?

Japanese word order is flexible, as long as particles stay attached to the right words. Some variants:

冬は暖かい帽子をかぶります。
Neutral, topic = winter. Very natural.
冬に暖かい帽子をかぶります。
Time marked by に; also natural.
暖かい帽子を冬にかぶります。
Still correct. Slightly more focus on 暖かい帽子 (what you wear), then specifying when.
暖かい帽子は冬にかぶります。
Also correct, but the topic becomes 暖かい帽子:
“As for warm hats, (I) wear them in winter.”
This sounds like you’re contrasting warm hats with other kinds of hats or times.

What you can’t do is drop or move particles incorrectly, e.g.
✗ 冬暖かい帽子をかぶります (without は or に) sounds unnatural in standard usage.

So yes, you can change the order, but the topic and emphasis change.

Why are there spaces between the Japanese words? I thought Japanese doesn’t use spaces.

You’re right: in normal Japanese writing, there are no spaces between words.
The sentence would normally appear as:

冬は暖かい帽子をかぶります。

Textbooks and teaching materials often add spaces (冬　は　暖かい　帽子　を　かぶります。) to help learners see word boundaries and particles more easily. It’s a teaching aid, not natural writing.