Domain names are the basic addresses of the internet. They have controls built into them the makes email possible, websites work, and more. Every website MUST have a domain name to function as without one, a user would be unable to get to your website files on the server.
Domain names are the URLs you type into your address bar of a browser. Think like facebook.com, google.com, reddit.com, etc. All of those addresses take a user to a specific server (or set of servers) that house the files and information of the actual website itself.
You can also change the domain name extension and get a different website. For instance, there could theoretically be a different website on facebook.org and facebook.com. You can actually see this with http://Wordpress.com and http://wordpress.org.
The extension of your domain name typically is supposed to give reference to what kind of organization you are. If you’re a business, you should likely have a .com extension. The .com extension stands for “commercial,” and the .org is usually supposed to be for a non-profit “organization.” If you’re trying to use your website to make money, or to represent a for-profit business, you probably want to just stick with the .com extension. Search Engines are also proven to give more merit to .com of others anyway.
Here's the thing...
If you want a website, the first thing you need to do is get a domain name. Domains are scarce in that there are only so many out there, and if someone else has the name you want, there’s no real way to get it from them. So, go get the domain you want ASAP.
Inside your domain name, you have different IP addresses built into it called the DNS. The DNS for a domain name controls where the user goes when visiting the website, as well as how email works, and more. Through an email client, you have the ability to change the MX records (mail exchange) of your domain name, giving you the firstname.lastname@example.org email addresses instead of being stuck with a standard gmail/yahoo/apple.com email extensions.
You can even look at using keywords of your business in the domain itself. If your business sells mountain climbing gear, and is called “Ascend Outfitters,” or something like that, you should first try and see if you could get ascendoutfitters.com. If not, you could take a look at ascendmountainclimbing.com or mountaingear.com or whatever variation you like. The shorter the better, and the more it represents your business the better.
Finally, it’s also wise to get misspellings of your name as well, and even possibly the .org/.net variations of it. You wouldn’t want a competitor buying up your other extensions and possibly taking that traffic from you.
Thinking about getting a new website or domain? Go get it. We use Godaddy because they’re so fast, but there’s other options like Network Solutions or Namecheap and others. It doesn’t really matter as long as you can get the domain you want.