The most common problem people have is that Java isn't installed. You can download the latest version of Java at http://java.sun.com. You will only ever need to do that once, and then you can use all Java programs including this one.
For other problems, it may be possible to see why they are not working by running it in a Shell or Console. You can do this by running either Copernicus.bat or the Copernicus shell script.
If you are still having problems, you can email the Tank Software help desk helpdesk@tanksoftware.com where someone will help you out. If you do this, try and include the error message(s) you are getting when you run the program.