You usually want to focus your brand-building efforts on a single domain name. Especially if you’re just starting to build an online presence for yourself or your organization, trying to build on multiple domain names can spread your efforts too thin.
However, it can be helpful to purchase additional domain names apart from your primary domain name. For example, if you have a .com, it’s usually a good idea to go ahead and buy the .org version if it is available (and vice versa). It’s also a good idea to buy common misspellings of your primary domain name, or other things people might accidentally type if they were looking for you.
Generally, when you buy these additional domain names, you don’t want to set up whole sites on them — you just want to set up a simple 301 Redirect to your primary domain. (Make sure you set up wildcard redirects so that links to any URL with the wrong domain name will end up in the right place.)