The success of Twitter has made people sit and take notice. Short form messaging has its uses. As a result, people have gone ahead and made their own open source and business-oriented Twitter-like applications. A short list of 10 notable Twitter-like applications:

Open Source Microblogging
Laconica http://laconi.ca/ which is the software which powers Identi.ca. Most robust of the lot. Use it if you want tour own Twitter.
Prologue http://en.blog.wordpress.com/2008/01/28/introducing-prologue/ - a Wordpress Theme which could be used as a microblogging platform.
Sweetter 2.0 http://sweetter.net/ - a Python-based microbloging service from Spain.
Jisko http://jisko.net/public- Affero GPL (AGPL) licensed framework for microblogging, PHP 5 and MySQL.Again, from Spain.
Yonkly http://yonkly.com/timeline - written in ASP.NET
Twoorl http://twoorl.com/ - a GPL (3) implementation of a microblogging service in Erlang using ErlyWeb.

Enterprsie Microblogging
Yammer http://www.yammer.com/ - Twitter for business
SocialCast http://www.socialcast.com/ - Friendfeed plus Twitter for the enterprise
I Did Work http://www.ididwork.com/demo - For project team so that they can send status messages.
Trillr https://trillr.coremedia.com/ - Another Twitter clone - it asks,"what's on your mind?"