The Windows package of Celestia is a self-extracting archive. Download it to your computer and then run it.
If you are running Linux, you should check first with your distribution. There is a good chance that the package is available to you in the format best suited to your installation. A pre-compiled 32-bit version is provided in the autopackage format. It should run on any distribution that has OpenGL and GTK+ 2.6.
Mac OS X
The Mac OS X package is a disk image. Download it to your computer, double click it, and follow the instructions in the README.
Celestia is an open-source project. As such, its source code is provided and is freely modifiable and redistributable as per the GNU Public License. Installation instructions are provided in the INSTALL file.