J2ME (Java 2 Micro Edition) is a feature that allows the device to run small, user-installable software applications written especially for mobile devices such as phones.
J2ME is a specific version of Java designed for phones. There are other versions designed for desktop and general-purpose applications. Some phones have a version of Java that is not J2ME, and therefore is not directly compatible with J2ME.
J2ME has faded in popularity since the advent of smartphones, which can run much richer, more powerful apps than J2ME allows.
Android smartphones are based on an advanced form of Java, but they are not compatible with J2ME.
J2ME applications can provide specific functions such as a tip calulator, they can be games, or they can be custom-written corporate applications. They can be Internet-enabled, so that, for example, a J2ME game might let you play someone else in real time over the Internet. They can be created by anyone - not just the phone manufacturer or your carrier.
You can download new applications to your phone at any time. Most phones allow you to download new applications directly to your phone using the phone's Internet browser.
There are several alternatives to J2ME. In the US, a major directly competing technology is BREW.