The books on the shelf are ordered by genre.

I placed an order for a new pair of shoes.

place an order 「注文する・発注する」 などという意味がある。Vのplaceは「置く」、Nのorder は「注文」という意味なので、 place an order で、 注文を置くというニュアンスで考えるといい。I'd like to place an order for office supplies. 事務用品の注文をしたいのですが。この場合、orderをVと捉えて、 I'd like to order for office supplies. と言うこともできるが、 place an orderを使ったときのほうが、 丁寧な響きになる。 また、place an orderは、make an orderに言い換えることも可。 Excuse me, I'd like to make an order. すみません、注文してもいいですか？

In order to succeed, you must work hard.

He was ordered home.

They study in order that they may enter the university.

I like to do things in order.

The project was executed by order of the CEO.

Her bookshelf is well-ordered.

The president issued an executive order to address the environmental issues.

The court issued an affiliation order to support the child born out of wedlock.