These are websites primarily for an individual's personal (rather than business) use, where the individual owns both the content and the domain (i.e. it isn't on or owned by a centralised platform). There are lots of benefits to individuals for doing this. Such sites are sometimes referred to as being part of the IndieWeb, the "small web" or the "non-commercial web".
Use the Add Site link at the top. There are two main options: Quick Add and Verified Add. The Quick Add lets anyone submit any site simply by entering the home page, although it needs to pass an approval process after submission before it is indexed. The Verified Add options can only be used by the site owner, who needs to verify that they own the site.
The easiest way to prove ownership is to use IndieAuth, but if you don't have that set up you can still submit your site with a Domain Control Validation (DCV) process similar to that which you may have used for other services, i.e. you upload a specific piece of content to the domain's root or TXT record. The Add Site link will guide you through the process.
Use the Add Site link at the top and choose one of the Verified Add options described above.
At the moment, sites added via the Quick Add are reindexed weekly, and Verified Add are reindexed twice weekly (verified owners can also trigger reindexes on demand). The Quick Add page has a summary of the differences.
The plan is for site owners to pay a small fee if they want additional benefits, such as greater indexing frequency, higher indexing page limit, ability to configure what is indexed, ability to trigger a reindex on demand, access to the API to add a search box to their site, etc.
This site has been developed by Michael I Lewis as a side-project.