The emoji we have on our phones these days come from Japan originally. Japan didn't use SMS even back in the 90s, they used MMS through an email gateway so they could fit more characters in there as Japanese characters use 2 bytes instead of one. Those MMS suported pictures and everything, so the phone companies included a set of smiley faces you could put into your messages and called them emoji (e moji, literally "picture character").
The problem was that all carriers had different emoji standards. So if you sent an MMS from a DoCoMo phone to a SoftBank phone and included an emoji, it didn't show up the way you wanted it.
This changed when the iPhone came along. Since the iPhone wasn't a carrier-branded phone it did not include the proprietary emojis the companies put on their own hardware, but the users wanted emoji, so Apple put in an emoji keyboard. That keyboard was enabled by default in Japanese iPhones but hidden everywhere else, but could be enabled if you knew how. People found out about it, liked it, and Apple reacted by enabling it anywhere. Android followed soon after.
There was still a problem with the encoding though, so in 2010 the Unicode Consortium decided that they should include emoji into Unicode so that they look the same everywhere, and they also included a bunch of new emoji for the "western market." Therefore, every Unicode-capable device with the appropriate fonts can display emoji these days, including regular computers.