Shiny is a term used to describe a Pokémon that is a different color than its usual design (but not because of things such as regional or gender differences). All Pokémon have had official Shiny variants since Generation II (Gold, Silver, and Crystal). The term "Shiny" didn't become official in its usage until Generation IV (Diamond, Pearl, Platinum, HeartGold, and SoulSilver). Prior to that, it was a fanmade term! The term originates from the fact that, when encountered, a Shiny Pokémon will have a "sparkling" effect and sound.
Shiny hunter is a term used to describe an individual who actively searches for Shiny Pokémon! It's more complicated than just encountering Pokémon aimlessly, though. Each game, including some spinoffs, have their own methods of increasing the odds of encountering a Pokémon (or the chance that you'll encounter one).
The methods of increasing Shiny odds varies from game to game. The odds for Shiny Pokémon have also increased dramatically since their introduction, making them easier and less time-consuming to find.