The International Standard Book Number (ISBN) is a unique reference, printed on the back cover of a book. It is used by publishers, booksellers, and libraries for ordering, listing, and stock control purposes, and has an impact on book distribution. Nielsen Book run the ISBN Agency for the UK and Ireland, but there are other agencies worldwide. US-based self-publishing providers, such as Kindle Direct Publishing (KDP), offer a free US ISBN for authors to assign to their books, but using this means that, for UK authors, their books will only be externally available to US book retailers and libraries. For UK authors wishing to have their book externally available to UK book retailers and libraries, a UK ISBN is required (not free).
How does a UK ISBN work? Every book assigned with a UK ISBN is held on the Nielsen database. When a customer orders a book through a UK bookshop or library, a search of the ISBN in the database will bring up the book's bibliographic data, allowing the book to be ordered directly from the publisher or distributor. (When self-publishing, the individual author is the publisher and distributor, and therefore responsible for providing the book to whomever has ordered it.)
How to obtain a UK ISBN. The quickest way is to buy an ISBN directly from the ISBN store. Registration is required, as well as a reasonable amount of form-filling! As at January 2020, a single ISBN costs £89.00. 10 cost £164.00. A unique ISBN is required for each type of published media, so a paperback and e-book requires two. In order to register to buy an ISBN, the author has to provide a publishing name (imprint). This can be a trading name or a personal name.
The pros and cons of a UK ISBN. With a UK ISBN, a book may be requested by a UK bookshop. Although pleasing to the self-published author, this may not be financially viable. Bookshops are geared to selling traditionally-published books; where distributors hold wholesale stock, readily discounted. Self-published authors, using a provider such as KDP, rely on a print-on-demand service: each time a customer buys a book online, the book is printed. There is no wholesale stock (but authors can purchase discounted copies to sell privately.) Since bookshops won't order a book online at the retail price, they ask the distributor (in this case, the self-published author) to supply the book, at a non-negotiable discount, which results in a tiny profit or, more likely, a loss, to the author. However, an advantage of assigning a UK ISBN is that the author has total publishing rights over their book. Where a US ISBN is assigned, the publishing rights belong to that provider, i.e, KDP. This makes it difficult for the author to take certain actions, such as remove a publication, if they don't have control over their book. And once an ISBN has been assigned, it cannot be changed. (Publishing rights are not the same as copyright. The author retains copyright of their work wherever it is published.)