Processing is a software & a programming language, designed for artists and visual thinkers.
What makes it useful to us is that it is free to download, very quick to learn, with lots of great documentation.
Processing is designed so that you can quickly see how your program works.
Processing 2.0 also exports to Android, so can be used to create Apps for mobile phones.
Here are some useful sites
The Processing website www.processing.org
The reference page that explains what each term in the Processing Programming Language means www.processing.org/reference
The learning section of the Processing site www.processing.org/learning
A great site with lots of examples of projects www.openprocessing.org
A great site with lots of video tutorials for beginners www.funprogramming.org