Essentials will not start if you are using an older version of Java. Essentials was built against Java 6, so when starting the server running on anything older you will get an 'UnsupportedClassVersionError' or 'Bad version number' error.
If you are using Java 4/1.4 or below you will be unable to start Bukkit at all, as Bukkit was built using Java 5.
While Bukkit only needs Java 5, Essentials and a large number of other plugins were built using Java 6, so you will be unable to start Essentials unless you update.
It is always recommended you install and use the latest build of a Java version, as the JVM can contain optimisations and bug fixes.
There are no known issues running Essentials with Java 7.
Updating Java on most operating systems is fairly standard practice, just follow the normal steps you would to update any program.
Updating Java on Windows is a simple task. You will need to go to the Java website and then download and install the latest version available on the site. You can find the link to the latest Java by searching Google for 'sun java', or clicking here.
Installing/Updating Java on Linux, is usually done by asking your package manager to do it for you. This varies distribution to distribution so if you don't know which package manager your system is using, it is best to google for a tutorial for your specific version.
Installing Java on older, unsupported versions of OSX can be a little difficult, but in most cases you will simply need to get the latest Apple Java release (OSX users on PowerPC look here), and then check the JVM preferences.
You can set the JVM priority by opening Applications/Utilities/Java Preferences.app and moving Java SE 6 to the top of the list. It should look like in this screenshot: