Equestria

The story of My Little Pony Friendship is Magic takes place in Equestria, a land inhabited by magical talking ponies and other sapient animals. Equestria is co-ruled by Princess Celestia and Princess Luna.

The founding of Equestria
The founding of Equestria has been originally described in the iPhone app Twilight Sparkle: Teacher for a Day. In the beginning there were pegasus, unicorn and earth ponies who did not live in harmony. Their hearts were cold as the blizzard that overtook their land. Each group of ponies decided to find a new place to live. But they all arrived at the same new place with the blizzard. The ponies figured out there was only one way to get rid of the blizzard, to treat one another warmly. The unicorns used magic to build Canterlot. Once the city was finished, they invited the pegasus and earth ponies to a party, which is now a yearly event called the Grand Galloping Gala.

Part of this explanation is included in the episode Hearth's Warming Eve.

Discord's reign of chaos
The series starts off with a prologue with narration about the princesses ruling Equestria, raising the sun and moon and maintaining harmony. Only in The Return of Harmony Part 1 is the time before their rule mentioned in the series. Princess Celestia tells Twilight Sparkle and her friends that before Princess Luna and herself stood up to Discord, he ruled over Equestria, keeping it in a state of unrest and unhappiness. Celestia goes on to describe that, seeing how miserable life was for earth ponies, unicorns, and pegasi alike, she and Luna discovered the Elements of Harmony and rose up against Discord, turning him to stone. Discord's spell is later broken because, as Celestia explains, "Luna and [herself] are no longer connected to the elements."

The regal unicorn sisters
The princesses are introduced in the prologue of the first episode, depicted in a series of medieval-like drawings with a narration that says "two regal sisters who ruled together and created harmony for all the land", and that "the eldest used her unicorn powers to raise the sun at dawn. The younger brought out the moon to begin the night."

Night Mare Moon
The narration continues: their subjects, the ponies, played in the day but "shunned" the night and slept through it, which made the younger unicorn grow bitter, eventually refusing to lower the moon to make way for the dawn. Her bitteress transformed her into a "wicked mare of darkness", Night Mare Moon. The elder sister reluctantly harnessed the power of the Elements of Harmony and banished her "in" the moon, taking responsibility for both sun and moon, maintaining harmony in Equestria. The events of the first and second episodes take place a thousand years after Night Mare Moon's imprisonment, upon which she is freed, but defeated again through the magic of the Elements of Harmony, only this time she is transformed back to her former self and returns to rule Equestria with her sister.

Ponyville
Ponyville is the main setting of the series. Ponyville is a small town that was founded by earth ponies, although over time pegasus ponies and unicorns have also made the town their residence.

Canterlot
Canterlot is home to the royal palace of Princess Celestia. It was also the home of Twilight Sparkle prior to her relocation to Ponyville. The name is a reference to Camelot, the home to King Arthur and his "Knights of The Round Table" as well as being a play on the word "canter," which is a three-beat gait possessed by all horses that is faster than a trot but slower than a gallop.

Everfree Forest
The Everfree Forest is located outside of Ponyville. It appears not to be bound by the same laws as the rest of Equestria since plants grow, animals care for themselves, and the weather changes without the help of ponies. It is also home to some strange creatures, including Ursas and parasprites.

Froggy Bottom Bog
Froggy Bottom Bog is featured in Feeling Pinkie Keen. Fluttershy relocates some frogs to the bog because of overpopulation. When Pinkie Pie gets an ominous premonition, Twilight Sparkle, Applejack, Spike and Pinkie Pie venture to the bog to make sure Fluttershy is okay. They come upon a hydra who chases them to a chasm which they cross in order to escape.

Appleloosa
Appleloosa is a Wild West town built by settlers about a year prior to the events of Over a Barrel. It is the home of Applejack's cousin Braeburn. The town is complete with the typical "wild west" buildings, such as a dance location, a saloon, and a sheriff. The settlers unknowingly planted an apple orchard on land traditionally used by a tribe of buffalo for their stampedes, leading to a heated territorial dispute. The name comes from the Appaloosa breed of horse.

Cloudsdale
Cloudsdale is a city in the sky where the Best Young Flyer competition takes place and where Equestria's weather is made. Only pegasus ponies live and work there, since non-pegasus ponies cannot stand on clouds. It is also the hometown of both Rainbow Dash and Fluttershy. The name is a reference to the Clydesdale, a breed of draft horses.

Fillydelphia
Fillydelphia is briefly mentioned by Princess Celestia in the episode Swarm of the Century. As an excuse for cancelling her visit to Ponyville, the Princess says she has to go to Fillydelphia in order to deal with an infestation of "incredibly bothersome creatures", presumably parasprites. Lauren Faust, the show's creator, mentioned in a comment on deviantArt that she wanted to name the main setting of the show Fillydelphia after Philadelphia where her family resides, but ultimately went with Ponyville.

Hoofington
Hoofington is a village mentioned by Trixie as the site of her alleged battle with an ursa. The suffix -ington is derived from old English for "town". The name is quite similar to that of a political news website, The Huffington Post, as well as the California town of Huntington Beach (aka Surf City, USA).

Manehattan
Manehattan is a large city where Applejack's aunt and uncle, the Orange couple, reside. Applejack is introduced to the high society of Manehattan in a flashback in The Cutie Mark Chronicles. She soon finds herself homesick, and returns to the farm. The name is a play on Manhattan Island, a section of New York City.

Trottingham
Mentioned in Stare Master, where Rarity is making 20 gold-lined gowns for ponies there, and later mentioned in Luna Eclipsed as Pipsqueak's previous residence. Seems to be a take on either Nottingham or Tottenham, a city in England and district of London respectively, and trotting being a walking speed for horses.

Whitetail Wood
A forest near Ponyville. Unlike Everfree Forest, it is completely under pony control. Tree sap is harvested there. It is also the site for the Running of the Leaves, a running event held by the ponies as shown in Fall Weather Friends to shake all of the leaves off the trees.

Ghastly Gorge
A gorge filled with perils, such as a windy cave, bramble bushes and quarray eels. Rainbow Dash held a contest to see who would be her pet there in May the Best Pet Win!

Trivia

 * Lauren Faust wrote in a comment on her deviantArt page that the Everfree Forest is west of Ponyville, Sweet Apple Acres is southwest, and Fluttershy's cottage and meadow are located north of the farm, presumably between Ponyville and the forest. Canterlot is northeast of Ponyville. However in subsequent interviews, both Faust and Jayson Thiessen mentioned that the geography and continuity of the show are loosely defined and may change as necessary.
 * ​The English word most similar to the placename Equestria is equestrian, "of or relating to horse riding", from the Latin equester, "belonging to a horseman", from equus, "horse", which is also the origin of equine. In this sense, "Equestria" would translate as "land of the horsemen" instead of "land of horses".