There are many websites that allow you to create your own website for free. The domain wouldn’t be your own unique domain unless you find the right hosting company. If you already have a website on a website hosting provider, usually they will offer a hosting package deal that allows you to own your own domain.
If this isn’t an option, choosing a sub-domain is also another option. Sub-domains are when you have a website with a host internet provider that allows you to keep your domain by attaching it to their domain. An example of this would be yourowndomain.webs.com if you went with Webs.com as your internet provider. Another example would be yourowndomain.wordpress.com, if you went with a blogging website.
Sub-domains are the easiest to obtain for free. Having a complete domain name usually costs money, whether if it is through a package deal with the hosting company or a monthly payment with the domain server. However, there are some services that allow you to set up your own free domain. These usually are referral servers. If you refer many people to a referral domain server, they usually will give you your own free domain.
There are some websites that list free domain names that can be used. This website offers a list of websites and domains that offer free domain names: http://www.freewebsiteproviders.com/free-domain-hosting.php.
When it comes to finding a free domain, you usually are looking at sub-domains or a referral service or at least some form of payment for a package deal with your current website host. It comes down to personal preference and what you are using your website for. If your website receives high traffic then your own domain might be needed for the higher bandwidth to endure the amount of traffic, preventing your site from crashing.