A purchase order is a written commitment from a customer to buy something, issued before the work happens and carrying a reference number.
You will meet them mostly on commercial work. Households do not issue them; property managers, contractors, councils and larger companies almost always do.
Purpose
An invoice without the purchase-order number does not get paid. It goes into a query queue, and nobody in their accounts department will chase you for the correction. It simply sits there.
This is one of the most common causes of a mysteriously slow-paying commercial customer, and it is entirely preventable.
It is your evidence of the agreement. If a dispute arises about scope or price, the purchase order is the document that settles it.
It caps what you can bill. Work beyond the purchase-order value usually needs a new one or an approved variation — see change orders. Doing it anyway is how contractors end up unpaid for work everybody agrees they did.
Routine
Ask for it before starting. Not at invoicing.
Put the number on the invoice, prominently.
Get a new one for extra work, in writing, before doing it.
Record it against the customer so it is not in somebody's inbox — see customers and vendors.
Issuing your own
You can also issue purchase orders to your suppliers, which is worth doing on larger material orders: it fixes the price and the quantity in writing before delivery, and it makes checking the invoice against the order possible.