Tiggerito - 12:25 pm on Mar 20, 2012 (gmt 0) [edited by: Robert_Charlton at 9:14 pm (utc) on Mar 20, 2012]
Any quality issue like broken links may be a negative signal to the search engines.
More importantly, it's a signal to visitors. No broken link (404 page) is a positive experience for a visitor and therefore they are less likely to continue with you.
Links from your website to other websites will just annoy them. No basic ranking issues there and at least they know you now and will remember your website is broken!
Broken internal links are easy to fix. So just do it.
Broken links to your website are harder to discover and fix. But they have the most value/cost.
They are links that are sending potential customers to your website. Broken and you've lost a lead before you even get a chance to talk to them.
They are links that add ranking to your website. Broken and you throw away that ranking.
If the linker finds they have a broken link on their website then they will most likely drop the link. If they are ranked themselves that's a big loss.
The first two are easy to fix. Use a crawler like xenu or screaming frog to find internal and outgoing link issues, and fix them.
For incoming links (backlinks) register your website with Google Webmaster Tools and check out the Diagnostics->Crawl Errors. Here you will see who has broken links to your website.
Ideally you want to contact them and ask them to update their links. This is not always possible and quite often fails. So the next best thing to do is to set-up 301 redirects to send those visitors to a page that works.
301 redirects often take a bit of tech knowledge. In most cases the website is hosted on Apache and uses php to run the website. In those cases you need to learn about the .htaccess file. It's always worth searching on your system plus "301 redirect" and you should find out how to do things.
You should be able to find all the things I mentioned via Google. If not, pipe up.
[edited by: Robert_Charlton at 9:14 pm (utc) on Mar 20, 2012]