Yes, you do need to register your chosen domain for it to become active on the web.
Once you've chosen your domain name, you can register it on the Dreamhost Panel here.
Depending on the TLD, some may require you to purchase registration for 1, 2 or more years at a time,
and yes, you would need to renew this registration if you wish to continue using the domain name.
During registration you would supply information for the registrar to supply to the registry, though you can (on certain TLDs) have the option of hiding your information.
Once you have registered your domain name, you can then add hosting if you haven't already done that bit first.