The history of the term Indigo Children and Star Children came out of the sseventies, the phenomenon appealing to parents whose children have been diagnosed with learning disabilities (particularly autism and ADD/ADHD) and to parents seeking to believe that their children are special (special snowflake syndrome), rather than seek out actual help.
The characteristics described are vague enough that many children could be considered an indigo child. Things like high intelligence, resistance to authority, empathetic and inutitive nature, strange to others, etc. This is called the Fouer effect.
Some believed that they are incarnated spirits or beings sent to usher the next age or are the next stage of human evolution. It's pretty much exclusively a new age belief.
Star Children are pretty much the same thing.
I, being someone who has aspergers, would have been considered a indigo child if I had lived in such an environment, so I really don't take it that seriously as a concept. Fluffy, maybe.