In the Web world, there is a kind of redirect called a 301 redirect. This kind of redirect tells search engines that the redirect is a permanent one - versus a 302 redirect, which is a temporary one. This kind of redirect is commonly set up on the web server - either in the hosting, with something called a .htaccess file or else in the code of the website.
What's happening is that the URL is being rewritten by the server or the code on the website and the website visitor is then redirected to another URL, which can either be on the same website / domain or another website. If you have used a URL shortening service to shorten a URL (website address), then you've personally set up some 301 redirects.
One common 301 redirect is to change a non-www URL to a www URL. For example, redirect someone from "yourdomain.com" to "www.yourdomain.com." This is done so that you don't appear to have 2 of the same/duplicate websites since a website with a "www" subdomain in front of it could go to another set of files for your website than the non-www version of your domain (if you so desired).
Another reason you may use a 301 redirect is if you change your domain name from one you have been using to a new one. When you use a 301 redirect, search engines know that there is a permanent change and doing it this way passes along all the "link juice" or credit (or history) that you had with the old domain. It's a great way to change your domain name and not lose any search engine rankings. You just have to make sure you use a 301 redirect (versus just a normal redirect or 302 redirect) and you should set it up in such a way where there's a redirect set up for each URL (every page) of your old website (match each old URL with a new URL).
You can also use a 301 redirect to get people to your website if they type in your domain name (website) wrong. For example, this website is webstix.com but that's not a common spelling for "sticks," so we also own websticks.com (try it - opens in a new window). We're redirecting people to webstix.com - not another whole website that has websticks.com in the address bar. That's the wrong way to do it and we see it a lot. We'll repeat this one more time...
You absolutely do not want the same website showing up under different domain names. This means that the same website shows up with different domain names in the website address bar in your browser. This is incorrect and you're only hurting your rankings. You need to use a redirect or else search engines will see the same content under multiple domains and then ding your website (penalize it) because your website contains the same content from other websites (duplicate content).