Keeping your site free of broken links will improve your user experience and also help your site rank better in search engine results.
Checking your site for broken links is very easy with a large number of tools to chose from.
Deadlinkchecker.com
Dead Link Checker is easy to use and it shows all of the links on the site and then it checks them. It seems to be one of the slower checks to run.
brokenlinkcheck.com
Broken Link Check does a great job of checking your site for broken links.
Clicking url takes you to the page itself, but as some links can be hard to find clicking src shows where in the page the link is. It's then simply a job to check the link and remove it. Sometimes it gives a false positive, like an example link I had in some code in an article.
Validator.w3.org
The w3 Validator is an "official" checker and it has no vested interest in selling you any commercial packages or services. The report isn't easy to use compared to the others.
Other Link Checkers
There are many other link checkers, including ones that run from your computer. Try searching Google for them.