In the Games:

Pokemon that are shiny are more commonly known in the games. If you search for Pokemon in wild grass, the sea and even in a cave, you might encounter a Pokemon that has colors not like what its species would have. For example, as shown below, a Shiny Pikachu is orange while a normal one is yellow.

It has been said that the chances of finding a different-colored Pokemon are 1 to 8192. It has also been stated that Pokemon Diamond and Pearl increased that chance by the usage of a Poke-Radar, a tool used to find rare Pokemon hiding in the grass, which makes the chances of encountering a shiny 1 to 40 by going into sparkling grass. If you have or find a shiny Pokemon, it'll sparkle at the beginning of a battle. In some cases, it'll be hard to determine a shiny Pokemon unless it sparkles. For instance, there isn't much of a difference with the normal and shiny sprites of a Persian:

If you look closely, the sprite on the right isA lighter than the one of the left. This prooves that shiny Pokemon will not always have different colors, just different color schemes that are either darker or lighter. One last thing, don't confuse a female Hippowdon as a shiny, it just has different color. So sorry if you thought it was shiny in the first place:

In the Anime:

Different-colored Pokemon have shown up in various episodes in the television series. Below is a list of encounters: Jackson from New Bark Town in the Johto region battled Ash's Cyndaquil with a darker-looking Magneton. A trainer named Jimmy, from Hoenn, accidentally caught a golden Magikarp while fishing. Possibly the most famous shiny Pokemon, a red Gyarados was caught by Lance the end its rampage from the Lake of Rage to Mahogany Town. Ash caught a Noctowl while traveling in Johto that had brighter feathers than that of a normal Noctowl. A Shuckle with a lavender shell was spotted when an old man saw it looking for berry juices. Ash, Brock, May, Max and Pikachu once achieved in paring a male Donphan with a pinkish female Donphan. During a gym battle against Winnona, Ash's Swellow battle another Swellow with green wings. There has been some rumors that Ash's Butterfree (in the episode Bye-Bye Butterfree) flew off in a swarm with another Butterfree that was shiny. Although it was pink its still not considered shiny. A shiny Butterfree has green eyes, a purple body, and wings with a shade of lavender.