Let's discuss it with example. if your previous website was www.xyzsite.com and you wanted to change it to www.goodxyzsite.com , you'd implement a 301 redirect from the old URL to the new one. Now whoever typed in your old URL (or clicked on a leftover link to your old URL), would automatically appear at your new URL.
Benefit of 301 is that it redirects are above all important from the point of view of Search Engine Optimisation (SEO). As it is the only server code that indicates the permanent move of a page to a new address, it is also the only code that is completely understood by search engines when indexing a site. Using a good server configuration, and well designed 301 redirects when releasing a new version of your website is thus crucial for your presence on search engines, and your potential incoming traffic.