From Bulbapedia, the community-driven Pokémon encyclopedia.
Misty's Togetic (Japanese: カスミのトゲチック Kasumi's Togechick) was found as a Pokémon egg by Ash Ketchum in Grandpa Canyon in Attack of the Prehistoric Pokémon. Ash, Misty, Brock and Meowth fought over the egg and competed to determine who would be the eventual owner. When the Pokémon Egg hatched into Togepi, the first thing it saw was Misty, and it chose her as it's trainer, thinking that she was its mother. It ended up being the sixth Pokémon Misty got in the Kanto region. Its voice actress as both a Togepi and Togetic was Satomi Kōrogi.
Although Togepi appeared harmless, it sometimes used Metronome to get out of dangerous situations. The first time it happened was in Pikachu Re-Volts. Butch and Cassidy's Drowzee was using Metronome to attack Ash and his friends. Misty's Togepi began using it, causing an explosion that sent Butch and Cassidy flying. It happened again in In the Pink. A Rhyhorn was about to charge the group when Togepi used Metronome to use Teleport. It also used the move later in the episode to stop a Hyper Beam attack by a Nidoking by using Reflect.
Despite the fact that Misty knew Togepi could battle, she only used it one time to battle Ash. In The Totodile Duel, Ash was using his Pikachu to battle Misty for ownership of a Totodile. Misty used Togepi, because she knew Pikachu would never attack it, as Pikachu would often babysit it. Using Charm, she ended up defeating Pikachu.
As Ash, Brock, May and Max met Misty near the gate to the Mirage Kingdom in The Princess and the Togepi, it was kidnapped again by Jessie, James and Meowth, who were temporarily working for Colonel Hansen, a crooked Mirage Kingdom official. Colonel Hansen wanted to get his hands on Misty's Togepi so he can rule the Mirage Kingdom. Since Colonel Hansen is an impure-hearted person, his taking over the Mirage Kingdom would destroy the Togepi Paradise. It later evolved into a Togetic in A Togepi Mirage!, and helped Ash and company defeat Colonel Hansen. Misty released her Togetic to protect the Mirage Kingdom and the Togepi Paradise and the all Togepi who lived there.
Rumor has it that if Togetic returns for the Generation III anime, it might evolve into a Togekiss.
Amongst many fans, Misty's obtaining Togepi is considered the moment when her personality "jumped the shark", and started to become drastically different. Prior to Togepi's introduction, Misty was known for being angry, active, and prone to fits of competitive spirit and rage. After she began to raise Togepi, her personality became significantly subdued as her character revolved more and more around the little Pokémon's wellbeing, leaving her little time for anything else.
Whether this drastic shift in character was caused in itself by Togepi or rather a writer's block by the producers over what to do with Misty in the Generation II series is unknown, but websites and organizations such as Pokemopolis will often point to her acquisition of the little creature as the sign of a sudden shift of much of Misty's character.
Perhaps in a bit of irony, a running joke on the Pokemopolis website is that Misty's Togepi was a creature from another dimension who wanted to take over a world and fed off of Misty's happiness. Bizarrely enough, much of that joke turned out to be (technically) true in the Mirage Kingdom mini-arc.
Before Misty hatched her Azurill, Togetic was the only Pokémon she owned that was not of the Template:Type2.
For more information on this Pokémon's species, see Togepi and Togetic.