CyanogenMod is a fork of Google's Android mobile operating system; it is free and the source is available as well. It provides ROMs for a large number of mobile devices and is extremely popular. It is now a commercial entity. (Disclaimer: the author has been using CyanogenMod on an LG Optimus P970 since July 2012.)
TextSecure is a product of Open WhisperSystems whose lead engineer is well-known open source hacker Moxie Marlinspike, author of the Firefox extension GoogleSharing (which anonymises data sent to Google while surfing) and lead engineer at Open Whisper Systems.
The company also has a product called RedPhone which provides end-to-end encryption for calls, so that nobody can listen in.
TextSecure ensures that all messages sent from one device running CyanogenMod to another are automatically be encrypted and secured. The same will occur in the case of anyone using TextSecure.
It will be incorporated into versions of CyanogenMod from 10.2 onwards. If the receiver is not using TextSecure or CyanogenMod, then the message would be unencrypted.