They are Pokemon with a different colour scheme to the ones normally encountered. For example, a shiny Golbat would be a Bronze-yellow instead of purple, and a shiny Aron would have red eyes instead of blue. Also, shinies emit sparks when they first enter the battlefield.
"Where do I catch shinies?"
Anywhere you encounter wild Pokemon. There are 3 possible types of Shiny pokemon you could encounter:
1) Wild shinies - these are the ones you will see while walking in an environment which generates Pokemon, i.e. grass, water (while Surfing) or in caves.
2) Egg shinies - An egg you recieve from the Daycare couple could be a shiny version of the egg's mother.
3) Event shinies* - These Pokemon can be obtained in a shiny version as well. Just before you engage one, save the game and then proceed to fight it. As soon as you see it is the normal colour, Soft Reset (press A + B + Start + Select all at once) your GBA/SP/DS/Micro and re-engage the Event Pokemon.
* Note: Event Pokemon are those you obtain by triggering a certain mechanism in the game. These include Starters and Legendaries.
"Where exactly will I find a shiny?"
You will never find a spot which yields a lot of shinies. They are randomly generated among the normal versions at all locations. Therefore, if you encounter a wild Wailmer in a certain pond, you will eventually come across a shiny Wailmer as well.
"What's the chance of me encountering a Shiny?"
This is the hard part. For every 8192 (2^13) Pokemon you encounter in the wild, 1 of them will be a shiny version. This ratio never, ever changes... Even if you have clocked 1,000+ hours on your save file, you will always have the same chance of finding a shiny.
"Will Shiny Pokemon still be Shiny after I evolve them?"
Yes. A shiny Pikachu will still be shiny, even after it has become a Raichu.
"If I breed a Shiny Pokemon, will the offspring be Shiny, or have a higher chance of being so?"
No, unfortunately. As a lot of the older gamers found out in Gold/Silver, the Red Gyarados doesn't have a Gold Magikarp as offspring. As stated, the chances of getting a shiny will always remain the same. I have never heard of a shiny Pokemon having shiny offspring... the chances of that happening are 2^13 x 2^13, which is 1 in 67,106,864. Ouch.
"Do Shiny Pokemon have better stats than the normal ones?"
It was a common-held belief that the above was true. Back in the Gold/Silver days, people noted that shiny Pokemon had higher IVs compared to their normal counterparts. This changed in Ruby/Sapphire - therefore, shiny Pokemon have the same range of IVs as normal ones.
"Are there any advantages to using a Shiny Pokemon in battle?"
Nope. The different colour scheme is superficial. It only affects the looks of a Pokemon, not the fighting performance.
"So... what's the point in having a Shiny?"
There's no real advantage in having a box full of shinies. They are seen as unusual, positive occurances which are unexpected. People usually catch shinies for bragging rights, or simply because they look nicer/fancier than the normal variables.
"Apart from the colour and sparks, is there any other way I can tell a shiny from a normal Pokemon?"
Shiny Pokemon have the same box icons as normal Pokemon, i.e. a shiny Golbat will have the normal purple icon, like a normal Golbat. However, if you look at the PokeDex number of a shiny, you will notice it is Gold coloured instead of white.
"Can I see shinies in Battle Tower?”
You can see shiny Pokemon in any randomly generated Pokemon, including Battle Tower and any other variants of in-game, non-field trainer battling. For example, someone on this forum encountered a Green , and I saw a Red Sandslash just the other day.
"Any strategies for finding a Shiny?"
Depends how you want to encounter one:
1) Breed - Simply keep obtaining and hatching eggs until you find the shiny version. This method is best used for rare Pokemon, like Ralts, Bagon or Feebas.
2) Soft Reset - for Event Pokemon only. If you want a Purple Kyogre or Black Rayquaza, use this method. The quickest method out of all of them, but still slow and painful. If it takes 30 seconds to attempt getting a shiny, you will theoretically spend 68 hours doing so.
3) Wild - simply run about in the grass where the Pokemon you want is usually. Eventually, you will encounter the shiny.
"So... Finding a Shiny in Ruby, Sapphire, Emerald, FireRed, LeafGreen, Silver, Gold or Crystal is exactly the same as how I can get them in Diamond/Pearl?"
Yes. There is no difference in the method of obtaining them. In fact, apart from the fact that Shiny Pokemon in the GSC generation had slightly higher stats, nothing has changed.
I hope this guide has helped those who needed to know about shinies. Good luck finding them.