You need a computer that can run at least Java 1.7 to use the full Jalview application (unfortunately this excludes users with older Macs - sorry). If java is installed, then the 'Launch Jalview Desktop' link at the top should work (if your browser asks you to save the application-x-jnlp file or choose an application to run it with, look for 'Java Web Start', or locate your java installation directory and look for the javaws binary). You can also go to the Install Anywhere Web Installer download area area to get an installer for your platform.
If you have Java 1.8 or later on in your browser (Firefox still provides this), you can try the web based version of Jalview - JalviewLite - right now by going to the applets page. The web based version doesn't let you print or access web services, but includes most of Jalview's core functionality.
Linux users: To launch jalview via webstart on openJDK from your browser, you will need to install icedtea-netx, which provides the javaws binary.