emoji are tiny images embedded in the phone browser. These images can be used as placeholders or icons to identify a link or other elements in your page. There are emoji of common use such as airplane or a train and there are some that may indicate emotions, sun, moon phases and more.
These icons are pre-installed in the browser and may be invoked with a simple syntax. No HTTP connection will be required. This can be useful to save bandwidth.
Emoji are referenced with an id. It is odd to see that i-mode devices sold in Japan use a different id for emoji as compared to i-mode devices sold in Europe. Some emoji are present in Japan and missing in Europe and viceversa. This makes it a bit inconsistent.
All you need to do to implement emoji on an i-mode phone is to place the appropriate character entity in your page. The browser will do the rest. If you wanted to display the emoji of the sun, you should place  in your page, if you are sending it to a Japanese device, or  if you are sending it to a European device.
Kyle Barrow provided the full list of emoji.
If you really like emoji and want to serve them to devices that are not i-mode you also read What is a pictogram?