DigitalOcean provides only hosting, it does not provide facility to purchase domain. So you have to get your domain name from the domain registrar. There are many domain registrar like goDaddy and hostingraja etc.
You can go with you own preference of domain registrar. Once you have purchase domain you to do some dns settings on both at domain registrar side and on DigitalOcean so your domain name can point to the ipaddress given by digital ocean to your droplet.j
In Simple manner you have to do some setup so we can bound your purchased domain name with your droplet ip address.
your_domain.com => DigitalOcean_droplet_ip
What we have to do is that just we have to change the Name Server of our purhcased domain name to Name Server provided by DigitalOcean.
If you have purchase domain from godaddy
Open You Domains Page
|Manage Domain godaddy
Click Manage DNS
In NAMESERVER card Click on Custom Name Server
|Change Name Server godaddy
Now Replace all the default name server with the name server of the DigitalOcean .
What ever domain registrar you are using at the end your name server should have only this name server in the list.
To check that you domain is transeferd to name server of the DigitalOcean use whois service.
I am using whois.net
|confirmation of name server for domain whois.net
Here you can check name server for you domain.
If it displays the name servers of digital ocean you done on registrar side.
Now Setup the DNS on digital ocean.
Login to you DigitalOcean account and click on add domain on DigitalOcean droplet
|Manage domain on DigitalOcean
Add you domain to DigitalOcean droplet
|Add domain on DigitalOcean
And done, it will setup the dns for your domain.
It will look something like,
|DNS Records DigitalOcean
you can add new record like CNAME for creating the alias or adding the sub domain or you can add MX records while using the mailing service from mailgun or other mailing serveries.
I will recommend to use virtual host so you can host multiple site with the same droplet. It is quite easy thing to setup the virtual host. you can find how to create virtual host here.
Hope you find it useful