Shiny Pokémon were originally introduced in Pokémon Gold and Pokémon Silver, and were incredibly rare - so rare that you'd be forgiven for not knowing they existed in 1999. With a 1/8,192 chance of seeing a shiny (1/4096 in the newer games) it's quite possible to beat a game multiple times and never see a shiny.
Shiny Pokémon are known for their different color (and they sometimes sparkle when they enter the battle). To obtain a shiny, you'll need a whole lot of luck. If you're lucky enough to have a sparkly wonder enter the battlefield, you'll then need to whittle down its HP before throwing a Pokéball, and pray you don't accidentally KO it (or hope it doesn't flee in Pokémon GO). In addition, if you happen to get a shiny Pokémon that enjoys fleeing from battle, such as an Abra or a legendary, you'll have to factor in either getting lucky with a turn 1 Pokéball throw, use your only Masterball (with a 100% catch rate) or have a Pokemon with a skill that prevents fleeing. There's also the problem of running out of Pokéballs or having your Pokémon faint.
Shiny hunting became a thing once players realized how involved it was to get an alternate color of their favorite Pokémon, and future Pokémon games started including various ways to increase the chance of a shiny, such as PokéRader chaining (fighting the same Pokémon over and over) and going to someone's Friend Safari in Pokémon X or Y.
A shiny Pikachu is an alternate colored Pikachu with a 1 in 4,096 of appearing in the wild in Pokémon GO and the newer, mainstream Pokémon games. These rates can be improved by various means, but catching one is still quite an accomplishment.
Unlike other Pokémon, the shiny variant of a Pikachu doesn't differ much from the original version.
The shiny Pikachu has gray-tipped ears and a bit of a darker tint (featured on the left.) The tails are the same, the only difference being that females tend to have curved tails.
The popularity of Pikachu means that if you're looking to collect a shiny variant, you'll probably get a chance either in Pokémon GO or the mainstream Pokémon games.
In fact, there's already been an event for Let's Go! Eevee and Let's Go! Pikachu to get a shiny colored Eevee or Pikachu, respectively, at Target.
On the other hand, if you want a Pikachu in Pokémon GO, be on the lookout for Community Events, such as the recent Pokémon movie event, where players could obtain a Pikachu wearing a detective hat by taking photos of the Pokémon they owned. Unfortunately, a shiny variant was not possible during that event.
To catch a shiny Pokémon on a normal day, numbers aren't on your side (one in thousands). That's why event Pokémon, normally obtained at a GameStop or participating game store, are your best bet. There's no nicknaming event Pokémon, unfortunately-- you'll be stuck with Pikachu.
To increase the rate of shinies, you'll have to refer to a guide- following certain steps like catching the same Pokémon over and over again can increase your rate of finding a shiny Pikachu, but even then, you're still looking at 1/100 or 1/64.
Finally, it's important to note that shiny variants of Pokémon evolve into shinies as well- if you give Pikachu a Thunder Stone (or evolve it in Pokémon Go,) you'll be looking at this beaut: a shiny Raichu.
Increasing Shinies by Chaining
In Pokémon GO, there isn't a known way to increase the shiny rate, aside from sponsored events that have a certain Pokémon show up much more frequently. The first method in mainstream Pokémon games to obtain a shiny is chaining.
Shiny chaining is not available in Pokémon Black, White, Black 2, White 2, HeartGold, or SoulSilver because there is no PokéRadar, which allows players to see spots in the grass where Pokémon hide. This radar is available after beating the Elite Four in Pokémon Diamond, Pearl, Platinum, X, Y, Omega Ruby, Alpha Sapphire, Sun, Moon, Ultra Sun and Ultra Moon. You can increase your odds by obtaining a Shiny Charm, but very few players will have the patience for this-- you need to complete the National Pokédex, meaning obtaining one of every Pokémon from generation 5 onwards. This increases the shiny rate by 300%.
Once you've obtained the PokéRadar, register it to one of your four directional buttons, and unregister all other items- using any one of them, such as a bike, will break the shiny chain. Throw on some Super Repels so normal Pokémon don't attack, as you're only looking to take on the patches of grass that shake, and use the radar to find the place your desired Pokémon will spawn. Head into the shaking grass to start the battle. If it's the one you want, great! If not, KO it and walk 50 steps to reset the radar.
Once you manage to find the Pokémon you want, KO it, and wait until you see a patch of grass that is shaking the same way as the first patch, at least 4 steps away from you. Go into that grass, find the same Pokémon and repeat the process. Every time you KO the same type of Pokémon, your shiny rate will increase, until you've KO'd 40 of the same Pokémon - that means it's at max.
Other Shiny Methods
In the Friend Safari, in Pokémon X and Y, Pokémon have a 1/512 chance of being shiny, but you'll have to add friends on your 3DS. Different friends mean different Pokémon in the safari- usually only a handful.
Breeding Pokémon from two different regions at the daycare (an easy way is seeing that one Pokémon has a name in a foreign language) will increase the rate of shinies, as will breeding with a foreign ditto. After this, it's a matter of running back and forth and hatching eggs, again and again.
Finally, fishing without messing up your streak will increase the rate of a shiny, as long as you don't ever let the Pokémon get away.
Shinies don't have better stats or any special abilities, and are there to give you a feeling of accomplishment, as well as bragging rights. Shinies are so popular, in fact, that some people actually sell them for a few dollars on Ebay, and will add you as a friend to send you the Pokémon once the purchase is completed. Catching a shiny Pikachu is no small feat, and sending one into battle against a friend, or leaving one at a gym in Pokémon GO is a great way to show off your hard work.