Slic3r can be downloaded directly from: http://slic3r.org/download.
Pre-compiled packages are available for Windows, Mac OS X and Linux. Windows and Linux users can choose between 32 and 64 bit versions to match their system.
The source code is available via GitHub: https://github.com/alexrj/Slic3r. For more details on building from source see § below.
Unzip the downloaded zip file to a folder of your choosing, there is no installer script. The resulting folder contains two executables:
slic3r.exe - starts the GUI version.
slic3r-console.exe - can be used from the command line.
The zip file may then be deleted.
Double-click the downloaded dmg file, an instance of Finder should open together with an icon of the Slic3r program. Navigate to the Applications directory and drag and drop the Slic3r icon into it. The dmg file may then be deleted.
Extract the archive to a folder of your choosing. Either:
Start Slic3r directly by running the Slic3r executable, found in the bin directory, or
Install Slic3r by running the do-install executable, also found in the bin folder.
The archive file may then be deleted.
For those wishing to live on the cutting edge, Slic3r can be compiled from the latest source files found on GitHub1.
Up-to-date instructions for compiling and running from source can be found on the Slic3r wiki.