Of course, Korea has a long history of its own music—traditional folk music in various forms. But Kpop, as we know it today, got its start as a result of the Korean War. Rock, Pop, and folk music from the West spread in the new 'South' Korea through the alliance of military servicemen. Koreans took the influence of this music and integrated it into their own. Many musical trends over the decades in the West were reflected in South Korea in their own way.
The 90s brought similar reflections of boy/girl-band pop along with hip-hop and techno. Singing and dancing in formation with elaborate choreography got it's start at the time. This is what I call the '1st Generation'. The Big Three companies were established and beginning to structure their training systems. Economically, a financial crisis in Asia affected South Korea in the late 90s. The first iconic idol groups surged in popularity for the escapism and fantasy they provided from the daily drudgery of school and work. They were a symbol of youth, wealth, and fame, which was a powerful motivator to the kids watching their TV sets at home. Many of those young fans would grow up to become the '2nd Generation' of idols.
The 2nd Generation made a huge leap in quality. They were selected for better looks, better voices, trained with better choreography, and many of the trainees worked for long years raising those standards for themselves. Costumes got flashier. Hair got bigger. The groups that debuted during this era gained rapid popularity and huge fanclubs built themselves into the culture in a big way. This is when companies were able to see the true potential of Kpop as a 'thing'—as a real power in South Korea. In early 2008, a big push could be seen. Branding of groups became paramount and the music industry was treated to a makeover in sound and style. The '3rd Generation' was born as the shiny, sleek, colorful, surreal, iconic explosion of groups and music videos we see today. Stylistically, there is a plateau currently. The 'look' of Kpop has become consistent in it's production value at the highest level. Trends shift in a more subtle way, usually pertaining to popularity of a certain sound (ie. jazz/brass/dubstep) or fashion concepts (ie. street-wear/overtly sexy/hyper cute).
On the current generation:
One could suggest that we are now in a 4th Generation. Characteristics might include the sheer massive volume of yearly debuts and global recognition beyond the fans that already exist. Many Kpop tours now include venues outside of Asia, including special festivals, concerts, and presence at pop-culture conventions. Whether Kpop will dramatically evolve again in the near future or fade into obscurity remains to be seen!