See Runtime.exec() and the Process class. In its simplest form:
Process myProcess = Runtime.getRuntime().exec(command);
...
Note that you also need to read the process' output (eg: myProcess.getInputStream()) -- or the process will hang on some systems. This can be highly confusing the first time, and should be included in any introduction to these APIs. See James P.'s response for an example.
You might also want to look into the new ProcessBuilder class, which makes it easier to change environment variables and to invoke subprocesses :
Process myProcess = new ProcessBuilder(command, arg).start();
...