Xcode is a set of tools created by Apple for the purpose of developing computer applications. Yes, you do need a Mac to use Xcode. It will not install on a Windows computer.
Xcode is mainly a development environment in which you would write your program's (or game's) code. You would use a language such as C++, Objective-C or Java. XCode does have interface-builders for laying out Application windows (such as which buttons go in the "File" and "Help" menus which are at the top of most programs), but almost everything else must be written in code.
To put things simply: Xcode is a program where you type your code, compile it and execute (run) it. It requires a Mac in order to run.
I am not certain how much (if any) programming experience you have, but if you would like to make a game, I recommend checking out Unity 3D (http://unity3d.com/). It has a free download, and you can follow the excellent tutorials on using it on this website (http://tornadotwins.com/).
Good luck with your game!
Also, for programming Android apps straight-up, I would recommend Eclipse (http://www.eclipse.org/).