zyugyoutyuu ha sensei no hanasi wo memosimasu.

授業中は makes during class the topic or setting of the sentence: As for during class, ...

You could also say 授業中に先生の話をメモします. That version is a more neutral time expression: I take notes on the teacher's talk during class.

So the nuance is roughly:

• 授業中は... = during class, ... / as for class time, ...
• 授業中に... = during class ...

With は, there can be a light feeling of contrast, like during class, this is what I do.

授業中 is made of:

• 授業 = class, lesson
• 中 = middle, during

Here 中 is read ちゅう and means in the middle of / during.

So 授業中 literally means in the middle of class, or more naturally, during class.

Japanese often leaves out the subject when it is obvious from context.

So 授業中は先生の話をメモします naturally means something like:

• I take notes on the teacher's talk during class
• or, depending on context, we take notes...

Japanese does this very often. If the speaker, listener, or situation already makes the subject clear, it is usually omitted.

Literally, 先生の話 is the teacher's talk or the teacher's speech.

In natural English, it usually means:

• what the teacher is saying
• the teacher's explanation
• the teacher's lecture/discussion

The word 話 can mean talk, speech, story, or what someone says, depending on context. In a classroom sentence like this, it most likely means the content of what the teacher is saying.

の links nouns together. It often works like 's or of in English.

So:

• 先生 = teacher
• 話 = talk
• 先生の話 = the teacher's talk / the talk of the teacher

But の is broader than English 's. It does not only show possession. It can also show relationship, description, category, and more.

Here it shows that the 話 is related to the 先生: it is the talk given by the teacher.

Because 話 is the thing being noted down.

In Xをメモします, X is what you make notes about. So:

• 先生の話をメモします = I take notes on the teacher's talk

Even though English uses take notes, Japanese often uses メモする as a verb-like expression meaning to make a note / take notes.

So the object marker を goes with the content being noted, not with メモ.

Yes. メモ is originally a noun, but in Japanese many nouns can combine with する to act like verbs.

So:

• メモ = memo, note
• メモする = to make a note / to take notes
• メモします = polite form of メモする

This is very common in Japanese, especially with loanwords and Sino-Japanese nouns.

You may also hear メモを取ります, which also means take notes. That is another very common way to say it.

メモします is the polite non-past form.

In Japanese, the non-past can mean:

• present/habitual: I take notes
• future: I will take notes

In this sentence, because of the classroom context, it most naturally sounds habitual or general:

• During class, I take notes on what the teacher says.

The basic structure is:

• 授業中は = topic/setting
• 先生の話を = object
• メモします = verb

So the sentence is built like:

[During class] [the teacher's talk] [take notes]

Japanese usually puts the main verb at the end, and modifiers come before the noun they modify. That is why 先生の comes before 話.

Yes, it is natural and understandable.

It gives the sense:

• During class, I take notes on the teacher's talk.

A few similar natural variations are:

• 授業中に先生の話をメモします。
• 授業中は先生の話をメモに取ります。
• 授業中は先生の話をメモします。

The original sentence is fine, especially if the speaker wants to emphasize during class as the setting or topic.