Emoji (pronounced eh−moh−jee) are iconic characters first introduced in Japanese cell phones in the early 2000s. At first different carriers each made up their own set of icons. NTT started off with a few. Then if I remember correctly J−Phone added many more than NTT to try to entice users to their system. Japanese, especially young Japanese women love using emoji and love to send messages to their friends full of them.
Eventually the various carriers agreed on a common set and relatively recently they were added to the Unicode standard which hopefully means all computers and all phones and tablets will be able to use them.
In the West there are or were several programs that insert "smileys". Many of those programs were probably malware but they were successful because even in the West many people like to insert at least faces like 😊 or 😞 emoji go far beyond just faces. A typical message might appear something like this
For whatever reason, lots of western tech people seem to hate this idea. They think emoji are frivolous and stupid. That might be true. What I'm not sure they realize though is just how big a part of Japanese culture they are at this point. They're built into pretty much all standard Japanese input systems.
In Japanese you enter words by sounding out the word using roman or hiragana characters and then choosing one of the words that sounds the same as what you just typed. For example you might type 't', 'o', 'r', 'u' and the system will display [とる」「取る」「撮る」「採る」「捕る」and others as some of the possible words that sound like 'toru'.
All of the major Japanese input systems now support emoji as one of your choices. Enter the Japanese pronunciation of the word for strawberry, "ichigo" and you'll not only get いちご、 一語、 苺 but you'll also get the emoji for strawberry (🍓).
To show what I mean, here's the Japanese input system on Mac (OSX) typing in various words and showing the options presented to the user.
Here's the same ones on iPhone.
And on Android.
And on Windows 8
What you should take away from this is emoji are not some "optional" thing Japanese people have to install or go out of their way to use. At this point they are fully integrated into their workflow. Any computer, phone or tablet will present these choices to them in their daily use which makes it super easy for them to use in their communications.
As far as browsers so far only Internet Explorer on Windows and Safari on iOS and OSX appear to support them. Support should be coming to other browsers asap. Texting from iPhone to iPhone seems to work. Email might work. If you read your email online and use a browser that supports it or you read your email in an Apple product you might be in luck.
Apple's support is fairly beautiful. Here are most of the emoji that are currently available as seen on OSX.
Windows 8 also supports them but they're not nearly as beautiful.
Outside of Japanese I don't know an easy way to insert them but if you want you can copy and paste from Wikipedia.
Sadly Facebook appears to block them. Given they're part of the standard Japanese input systems (see above) it seems kind if arbitrary to not down right culturally insensitive to delete some of the characters all Japanese systems let you enter. If you write software that doesn't currently support these characters I hope you'll seriously consider updating it to support emoji.