domain name registrar

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A domain name registrar is an organization that will register domain names for you typically for an annual fee.

Criteria

Configurability - You should be able to set up all record types (A, CNAME, MX, TXT) for as many subdomains as you'd like/need.
Email forwarding - does the registrar offer free email forwarding? This is useful if you want an @yourdomain.com email address that forwards to an existing account.
Auto-renew - it's easy to accidentally let a domain name registration expire; some registrars let you set up email reminders or an automatic renewal to prevent this from happening.
Domain Privacy - Note that most domain name registrars will make your personal information (name, mailing address, phone number and email address) publicly available via whois lookups. Some registrars offer domain privacy options, so that instead of your personal details the registrar's details will be in the whois directory. Only use domain privacy if you fully trust the provider of the service -- disputes about domain name administration or transfers may get tricky if you are not listed as the legal owner of the domain.
Bundled hosting? Shop around for the best registrar and best host for your needs. They do not need to be the same.

Registrars that IndieWeb community members use (or have used at some point)

Directnic.com
gandi.net
hover.com
Name.com
- optional Whois Privacy Protection for $4.99/year (as of 2017-02-10)
NameCheap
- Free Whois privacy protection (WhoisGuard) [1]
Dreamhost
- Free whois privacy
Domain Monster
Google Domains
DNSimple
OVH
Nomeo
GoDaddy
- Faced criticism and boycotts in 2011 for their support of US anti-piracy legislation, SOPA, though their official position was later reversed.
- Wikipedia moved off of GoDaddy in early 2012, partly because of this.
- Under a new CEO, GoDaddy has been in support of recent net neutrality regulations
eu.org (free!)

Kartik Prabhu: uses NameCheap...
Aaron Parecki uses NameCheap, dreamhost.com, and gandi.net, depending on TLD availability and price.
Kyle Mahan: uses name.com (as a registrar). Good prices, good UX, highly configurable.
rascul uses gandi.net. Great service, often slightly more expensive than NameCheap but is (or at least was) the best price for .io TLD.
Will Norris: uses Google Domains. As of 2014-06 it is still in limited beta, but I highly recommend it.
Bear: uses Hover.com.
gRegor Morrill uses Domain Monster. Good prices, mediocre UX.
Hans Gerwitz uses Gandi.net because the UI is simple and they are an ethical, European company
Arne Govaerts uses Nomeo. Small Belgian register with OK prices and good support. Free .be or .nl domain for univerity or hogeschool/hochschule students.
η uses eu.org, because it was free :)
...