BabyTwit!

Friendica is a distributed social network application which runs on commodity hosting platforms - PHP/MySQL/Apache. (If you can install Drupal or Wordpress you can probably install Friendica)

GNU social is a continuation of the StatusNet project. It is social communication software for both public and private communications. It is widely supported and has a large userbase. It is already used by the Free Software Foundation, and Richard Stallman himself.

Ello is a simple, beautiful, and ad-free social network created by a small group of artists and designers.

Plurk is a social journal for your life.

Plerb is a social microblogging and chat community.

Chttr.co is a social network with 1. openness — everything is public 2. effortless discussions 3. sensible line on free speech.