Most people don't need Java. Java is a technology that allows you to do things with a website. Kind of like Flash lets you play Club Penguin or view YouTube videos and Microsoft Silverlight lets you watch Netflix videos on your computer.
Unless you have some kind of online game or online software that needs Java, it's probably fine to disable it. Just right click the icon and find the menu selection that allows you turn off update notifications. Next step is to go to your browser (IE?) and navigate to the BHO (Browser Helper Object) section and disable Java.
To turn off Java in IE, just select Tools > Manage Add-Ons
If you should ever need Java while online, a pop-up at the top of your browser will alert you that Java is needed. You can choose to enable Java if your experience would be improved on that website. But more likely, if your experience is fine without Java, you can choose to ignore the warning and leave Java alone in it's disabled state.
It's ok to update to the latest version of Java, it's probably best to do that. But it's probably even better to do that and then turn off Java. If after a month or so your computing experience doesn't seem to have been affected, you could probably consider removing Java altogether.