Where broken links come from
Nobody creates a broken link. They appear when something else changes. A page was renamed. A service was dropped and its page deleted. The site moved platforms and every address changed shape. A blog post linked to another site that has since gone. An old flyer or directory listing still points at an address from three redesigns ago.
Each one sends a visitor, or Google, to a page that says "not found". The visitor leaves. Google notes that the page it had indexed is gone, and any ranking it had drains away. Individually these are small. On an inherited site there are usually dozens, and together they are a slow leak of both traffic and trust.
Finding them
Three sources, and you want all three.
Google Search Console. The Pages report lists URLs Google tried to reach and could not, under "Not found (404)". This is the most important list because it is the one Google is actually hitting, and it includes old addresses from before a redesign that you would never think to check.
A crawl of your own site. A crawler follows every link on every page and reports the ones that fail. Free tools exist for small sites; a developer will use a proper one. This catches internal links to pages that were deleted or renamed.
Your own server logs or analytics. Which 404 pages are actually being requested by real people, and from where. A 404 that gets a hundred visits a month from an old directory listing matters more than one nobody visits.
Combine them into one list: the broken address, how many visits it gets, and what it used to be.
The three-way decision
For each broken address, one of three things is right.
Redirect. The page moved or was replaced by something equivalent. Send the old address to the new one with a permanent redirect. The visitor lands on the right page, and Google transfers the old page's standing to the new one. This is the answer for renamed pages, restructured URLs, and dropped services that have a close equivalent.
Redirect to the parent. The page is genuinely gone and nothing replaces it, but there is a sensible broader page. An old post about a discontinued treatment redirects to the treatments page. Better than a 404, honest enough not to confuse.
Leave it as a 404. The content was junk, spam, or a test page; or it is being requested by bots and nobody real. A clean 404 is the correct answer, and redirecting everything to the home page is worse than a 404, because it tells Google the home page is about everything and nothing.
The rule: redirect to the closest equivalent. Never redirect to the home page just to make the number go away.
Setting redirects that stick
Permanent, not temporary. A 301 redirect tells Google the move is final and to transfer standing. A 302 says it is temporary and to keep the old address. For anything that has actually moved, it is 301.
One hop. Old address to new address directly. Not old to older to newer. Redirect chains slow the visitor and dilute the transfer.
At the platform or server level, not in a plugin if avoidable. A redirect rule in the server or hosting configuration survives theme changes and plugin cleanups. A plugin-managed list is acceptable if it is documented, but it is one more thing to break.
Written down. A single file or table of every redirect, with the reason and date. When the site is next rebuilt, this list is what stops all the old addresses breaking again.
The external links
Links from your pages to other sites break when those sites change. Less urgent for your rankings, but a visitor clicking a dead link on your page blames you. Fix by updating the link to the new address if there is one, or removing it. A crawler finds these too.
The special case: after a redesign
Every redesign that changes addresses needs a redirect map made before launch, old address to new, for every page that existed. It is the single most skipped step in website rebuilds and the single largest cause of "we rebuilt the site and traffic dropped". If you are inheriting a site that was redesigned without one, the Search Console 404 list is where you reconstruct it.
Where this sits
Finding the 404s and setting the redirects is a standard item in the fix half of our diagnosis, and a monthly glance at the Search Console errors is part of a care plan, because new ones appear whenever anything changes. It is unglamorous, and it is one of the few SEO tasks with a guaranteed return.