301 redirects are the best method to give a new destination to an obsolete URL. But it's not the only thing you can do with them. Here are three potential use cases.
1° the one we've just mentioned, i.e. redirecting an obsolete URL (to avoid a deadly 404 error) to a new active page. You can do this either from a URL on a given domain to another URL on the same domain or between different domains, if you have for instance changed the brand of your service and want to permanently redirect the old pages to the new presentation.
2° point a URL on your domain to an external service. Imagine for instance that you're using a SaaS platform for a contest but don't have the possibility to connect your project to a custom domain, you can create a URL like: mydomain.com/contest which will redirect to the service. That way you're using your own branding instead of URL shortener (bitly & co) or the long complex URL of the third-party service.
I'm also using 301 redirects to point to a PDF download URL ultimately handled by Delightful Downloads. A URL like mydomain.com/catalogue is much easier to remember than mydomain.com/?ddownload=1234 ( which is the type of URL you'll get from Delightful Downloads).
3° I'm also using 301 redirects when I want to create a URL to track the traffic from a specific channel. For instance the URL on my Instagram profile is a 301 redirect to my homepage. Much better in terms of branding than a Bitly URL, even customised.
The cool thing is that for both 2° and 3° I can see the stats of the traffic going through my 301 redirect via the plugin I'm using for this purpose: EPS 301. Over the last 5 days, I've had 42 visits from instagram ;-) Nice start.