Originally, Derpy (commonly referred to as "Derpy Hooves") first appeared in the first half of the pilot episode, "Friendship is Magic: Part 1" where she was seen in the background at Twilight Sparkle's surprise party. Notably different from the other background ponies, she had her eyes crossed in an animation error and her mouth was scrunched up in an odd manner. She was nicknamed "Derpy" by the fans after the slang term "Derp", which implied an often briefly absent minded behavior or careless mistake, often associated with eyes not aligned correctly or crossed. Needless to say the animators of the show caught on to this and ran with it.
Often used as a hidden "Where's Waldo" type Easter eggs in many episodes, Derpy Hooves has evolved into one of the most popular characters. She's often been given subtle nods in many "My Little Pony" types media outside of the show, including trading cards, a SDCC exclusive figure, and most recently the comic series being produced by IDW Publishing.
Outside of that anything about the character is usually always fan created. What's known of her is that she's a mail pony, has a bubble patterned cutie mark and has a taste for muffins. She also seen frequently paired up with Time Tuner (aka Dr. Whooves) though this isn't really stated by the show runners. A running fan gag with her is that she often does the opposite of a task given to her. For example, if asked to fix up a building, she'll demolish it, while asking to destroy it will result in her building it. This is unintentional on her part of course as Derpy's only trying to help.
Friendship is Magic: Part 1
Derpy Hooves makes her first appearance. Most notably in the background of the surprise party Pinkie Pie threw, where her signature "derp face" caught fan's attention.
Winter Wrap Up
Catching the attention of the show's producer, Lauren Faust, a similarly identical pegasus named "Ditzy Doo" played a small part, failing to go lead the southern birds back for the spring, instead flying north.
Feeling Pinkie Keen
In a brief scene, Twilight Sparkle ignores Pinkie Pie's ominous pre-cognitive senses that something would fall out of the sky. In return, she has a number of large, clearly heavy objects like pianos and anvils falling on her. This was then shown to have been Derpy's fault, as she was delivering those objects on a truck with other pegasus ponies, and accidentally dropped the load. This was the only time Derpy was ever seen delivering anything, probably because she was fired. However it was a nod to the fan theory that the bubbles on her cutie mark represent bubble wrap.
The Last Roundup
In the opening scene of this episode, Derpy Hooves finally gets her first speaking role. Trying to help Rainbow Dash set up decorations for Applejack's farewell as she was going to be sent off for a rodeo competition, Derpy inadvertently ruins town hall instead. Flying backwards into support beam and falling through a wooden floor as Rainbow Dash tries to keep her still, she lives up to her fan supported ditzy behavior.
Unfortunately, this scene was edited out due to a heated controversy over her name being "Derpy"(who some people claim was offending to the mentally disabled though that was never the intention), her signature eyes were even realigned and new lines were re-recorded in reruns of the episode. She was also renamed Ditzy Doo.
Despite this very unpopular and unnecessary use of censorship, Derpy Hooves still appeared as her old cross eyed self anyways. Theories have also been speculated how some cameos she's had could even be possible, appearing in flashbacks, as a younger filly and even inside a snow globe in just some of her appearances.
She likewise makes a few more appearances in the comics though usually either as background or brief cameos. In Issue 1 (of which she also appeared in at least 4 alternate covers). She's captured by the changelings and her form stolen by one of them (humorously getting the eyes wrong until a fellow changeling bonks him in the face, fixing it) while the Mane 6 and Spike look on. It's assumed she was rescued along with the town.
In Rarity's micro-story, she's seen passing over the spa while doing her mail delivery duties, scared off by the eager Rarity and hippie spa workers who try to sell her a beauty product.