Learn Hardcore Japanese: watasi ha tukue no ue ni hon wo narabemasu.

が is mainly a subject marker and tends to introduce new or emphasized information. You might use が if you were stressing that I (not someone else) am the one who arranges the books:

私が机の上に本を並べます。
I am the one who lines up the books on the desk. (emphasis on I)

But in a neutral sentence where you just state what you do, 私は is more natural.

What does の do in 机の上?

の connects two nouns and often means “of” or shows a possessive / belonging relationship.

机 = desk
上 = top / above
机の上 = the top of the desk / on top of the desk

So 机の上 literally means “the desk’s top” or “top of the desk”.
This pattern is very common:

日本の本 = a book of Japan → Japanese book
学校の先生 = a teacher of the school → school teacher

What does に mean in 机の上に, and why is it needed?

Here, に is a location/goal particle. It marks the place where something ends up or where a state exists.

机の上に本を並べます。
→ You are arranging the books onto / on the top of the desk.

The structure is:

机の上 = the top of the desk (a location)
机の上に = on / onto the top of the desk (location + particle)

Without に, the location would not be properly marked and the sentence would be ungrammatical.
In this sentence, に focuses on the final position of the books: they will be lined up on that surface.

Could we use で instead of に (机の上で本を並べます)? What’s the difference?

Both are possible, but there is a nuance:

机の上に本を並べます。
→ Focus on the resulting location: the books end up arranged on top of the desk.
机の上で本を並べます。
→ Focus on the place where the action happens: you perform the action of arranging on the desk.

In this particular verb, where the action naturally results in the books being placed somewhere, に is very typical and often sounds more natural.
で is more about “doing something at/in a place”; に here is more like “placing/lining them up onto that place”.

Could we just say 机に本を並べます instead of 机の上に本を並べます?

You can say 机に本を並べます, but the nuance is different.

机の上に本を並べます。
→ Clearly means “on top of the desk” (the surface).
机に本を並べます。
→ Means “arrange books at/on the desk”, but it’s less specific; it doesn’t explicitly mention the top surface.
Depending on context, it might sound a bit vague or slightly off, because we usually picture books on top of the desk, not inside it or at its side.

If you want to be clear about on the surface, 机の上に is more natural.

Why is 本 followed by を?

を is the object marker. It marks what the verb directly acts on.

本 = book / books
本を = book(s) as the direct object

The verb 並べます (to line up / arrange) is a transitive verb and needs an object:
何を並べますか？ → What do you line up?

Answer: 本を並べます。 → I line up books.

So 本を tells us that the books are what is being arranged.

What exactly does 並べます mean, and how is it different from 置きます?

並べます is the polite form of 並べる, which means:

to line up,
to arrange in a row,
to set out in order.

It implies order and often a linear or organized layout.

置きます (from 置く) just means:

to put,
to place,
to set something down.

Compare:

机の上に本を置きます。
→ I put a book / books on the desk. (no implication of order)
机の上に本を並べます。
→ I arrange / line up books on the desk (neatly, in some order).

What is the dictionary form of 並べます, and how is it different from 並びます?

並べます → polite form of 並べる (dictionary/plain form).
並びます → polite form of 並ぶ.

These are a transitive / intransitive pair:

並べる (transitive) – to line something up
- Needs an object: 本を並べる (line up books).
並ぶ (intransitive) – to be lined up / to line up (by itself or by people)
- No direct object: 本が並ぶ (books are lined up);
  人が並ぶ (people line up / stand in a line).

So in 本を並べます, you must use the transitive verb 並べる, not 並ぶ.

What tense/politeness level is 並べます? How would I say past, negative, or a question?

並べます is:

Polite (ます-form)
Non-past (can mean present or future, depending on context)

So 並べます can mean:

I line up (books on the desk) [habitually / generally].
I will line up (books on the desk).

Common variations:

Past (polite):
- 並べました → I lined them up.
Negative (polite):
- 並べません → I do not / will not line them up.
Past negative (polite):
- 並べませんでした → I did not line them up.
Question (polite):
- 並べますか。 → Do you / Will you line them up?

Do we really need 私? Can we omit it?

私 can usually be omitted if it is clear from context who the subject is.

(私は)机の上に本を並べます。

In natural Japanese, this would often be said simply as:

机の上に本を並べます。

We include 私 when:

We need to clarify who is doing the action.
We want to contrast with someone else (e.g., I do it, not you).
We are at the start of a conversation and setting up the topic.

Otherwise, omitting 私 is perfectly normal and often sounds more natural.

How flexible is the word order in this sentence? Can I move parts around?

Japanese word order is more flexible than English because particles show each word’s role.

Basic version:

私は机の上に本を並べます。

Some other acceptable orders:

私は本を机の上に並べます。
机の上に私は本を並べます。
本を私は机の上に並べます。

All of these still mean essentially the same thing: you arrange books on top of the desk.

However:

The verb almost always comes at the end.
Moving phrases changes emphasis / focus slightly, but the core meaning stays.

You cannot drop the particles (は, に, を) when you shuffle the order.

How do we know whether 本 means “book” or “books,” and where are “a” / “the” in this sentence?

Japanese:

Usually does not mark singular vs plural for common nouns.
Has no articles like a / an / the.

So 本 can mean:

book
books

and 机の上に本を並べます。 could be:

I arrange *books on the desk.*
I arrange *the books on the desk.*
I arrange *some books on the desk.*

Context decides which is natural.

If you need to be specific, you add more information:

一冊の本を並べます。 – I line up one book.
何冊かの本を並べます。 – I line up some books.
その本を並べます。 – I line up that / those book(s).

But in the original sentence, 本 is just “book(s)” without specifying number or definiteness; the listener uses context to interpret it.