According to the online documentation Titanium officially supports these operating systems: Windows 7, Mac OS X 10.6 or later and Ubuntu 9.10 (both 32bit and 64 bit versions).
However, the relatively low resource requirements and cross-platform nature of the tools and development kits means, in practice, you can use Windows Vista and (probably) XP and a variety of Linux platforms.
Go to the product downloads page to obtain a Titanium Developer Tool installation file for your platform, which invokes a sign-up step to add yourself to the Appcelerator Community. The installation process includes accepting the Apache 2.0 licence agreement.
An investigation of the target installation folder reveals a collection of executable and web page script resources. And a quick look at the desktop confirms we have a new Titanium Desktop icon.
Appcelerator provide the Eclipse-based Aptana Studio. This excellent IDE has all the functionality you'll need to create, edit, build and debug your apps. It's also perfect for general web development activities.
The Titanium toolset is the only software you will need to create desktop applications. However, mobile apps require some additional steps, something I'll discuss later.