A citation is any place on the internet, other than your own website, where your business name, address and phone number appear together. A directory listing is a citation. So is a trade association page, a chamber of commerce listing, or a local news piece that prints your number.
Why they exist at all
Google cannot ring you up to check you are real. It works out whether a local business exists, and where, by looking at how many other sources say the same thing about you. Twenty sites all showing the same name, the same street and the same number is a strong signal. Two sites disagreeing is a weak one.
That is the whole idea. A citation is not a link and does not need to be one. The value is in the details matching.
Which ones are worth having
Start with the big general ones: your Google Business Profile, Bing Places, Apple Business Connect, Yell and Facebook. Then the ones for your trade, such as Checkatrade, TrustMark or a manufacturer approved installer list. Then anything genuinely local, like the town council business directory or the trade association members list your membership number already entitles you to.
Ignore anyone selling you five hundred citations for fifty pounds. Those are auto generated listings on sites nobody visits, and they tend to introduce the wrong number or an old address, which is the one thing you are trying to avoid.
A worked example
A joinery firm in Derby changes its mobile to a landline. The new number goes on the van, the website and Google. Nobody touches the Yell listing, the Checkatrade profile or the old chamber of commerce page, so three sites still show the mobile.
Six months later the mobile is cancelled. Anyone who finds the firm through those three listings rings a dead number and calls the next firm on the list. Nothing looks broken from the inside, which is why this goes unnoticed for years.