301 Redirect is Permanent Redirection which means your server tells Search Engines that the link has moved permanently and they need to update their Database with new URL.

Whereas 302 is Temporary which means the redirection could be reverted back to original in near future or the link could be deleted completely.

Always use 301. Google Loves that... so its good for your site.

301- Permanent redirect
302 - Temporary redirect..

301 is a great way to redirect your break links. officially this most suitable for use. I always like to use it.

Well, here is something I've always wondered. If another web site has a broken link to your site, and you do a 301 redirect, does google continue to index that broken link? Does it just index it as <broken link> but deep down it knows it is truly a link to <fixed link>, and we just don't have to worry about it again even if that broken link lasts for a long time?

Alternatively, if there is a broken link, and google determines that it results in a 404 on your page is that final, or is it never too late to provide that redirect?

And, if a link is broken, does that mean it's only a matter of time before Google drops (or ignores I guess) that link outright?

'301 google loves that' is a bit overstated to say the least.

As already explained what they mean you should not always just use one or the other. You need to understand the difference and use the correct one.

If something is moved temporarily use 302.
If something is moved forever use 301.

That simple.