How do I set up a custom domain for my blog?

Pre-login steps:

Choose a registrar, or use tools like Yahoo Domains or Domainr to find and buy a domain name that's available. (This usually costs about $10-$40 per year.)
Make a configuration change depending on the number of levels in your desired domain. Your registrar should have instructions on how to do this.

Domain	Configuration
Two Levels (e.g. mywebsite.com)	point A-record (IP address) to 66.6.44.4
Three or More Levels (e.g. www.mywebsite.com or blog.mywebsite.com)	point CNAME record to "domains.tumblr.com"

Post-login steps:

Click Settings (the gear icon) at the top of your Dashboard.
Click the blog you’d like to update on the right side of the page.
Click the pencil to the right of the username section and enable "Use a custom domain."
Enter your domain (e.g. mywebsite.com) or subdomain (e.g. blog.mywebsite.com).
Click “Test your domain.”
Correct problems if the test finds any, and click “Test your domain” until the test is successful.
Hit “Save.”

Do I need to change the nameservers for my domain?

Nope, just the A-record or CNAME record depending on levels in your domain.

How do I know if I set up my domain properly?

After re-configuring your domain, you must wait up to 72 hours for the changes to take effect. When you visit the subdomain or domain, you should see a Tumblr error page — this means that the domain is correctly pointing to Tumblr, but that your blog hasn’t been configured to use it yet.