Equestria is a peaceful land inhabited by sentient ponies. Quarrelling and conflict are quite rare, and actual violence is even rarer.
There are four confirmed types of ponies in Equestria:
Earth ponies are known for their connection to the land, being able to grow food and cook it with better quality than other ponies. They are also known for being slightly stronger as well.
Pegasi can fly, and they can handle clouds like they were solid objects. The majority of pegasi help control Equestria's weather, and a lot of them also work as couriers due being able to cover more ground in a day than other ponies.
Unicorns have a horn on their head which allows for the use of magic. The first bit of magic a unicorn will learn is usually a telekinesis spell. You can tell if unicorn magic is being used, as their horn and all affected objects will be surrounded with a visible magical aura.
In game terms, all unicorn magic has Noticeable: Aura glows around horn and affected objects.
And finally, there are alicorns. They are immediately recognisable as such due to having both a pegasus' wings and an unicorn's horn with enhanced magical ability. All four known alicorns are also considered Princesses, and they are all considered to have earned their title. In short, if you should ever encounter one, show respect!
Something of note is that any alicorn can declare any non-pony a "Friend of Equestria". A Friend has to actually do something to help Equestria to be allowed the honour. Simon "The Griffin" Wentworth and Cass "Ms. Magus" Valentine are known to be the first two Friends to come from Earth, both becoming Friends the hard way, by protecting the first six ponies to come to Earth from mortal danger.
What the Griffin and Ms. Magus don't know yet, is that the title also means that ponies in a foreign land tend to want a local Friend as a guide if the area has any who are willing, and otherwise tend to defer to one when trying to understand a local situation.
Princess Celestia: Ruler of Equestria. Princess Celestia's job is to move the sun and watch over Equestria during the day. She is powerful enough that everypony will swear their oaths in her name. For example: 'As Celestia is my witness!'
Princess Luna: Princess Celestia's sister. Her job is to raise the moon and watch over Equestria at night. She was once so feared, that Equestria's version of Halloween was named after her name at the time, Nightmare Moon. She also has the ability to enter someone's dreams, usually to offer much-needed advice.
Canterlot (CC, GC): The capital and physical centre of Equestria. The most notable building is Canterlot Castle, the official residence of Princesses Celestia and Luna. It also stores the Elements of Harmony, the six most powerful magical artefacts in all of Equestria. Only a pony who embodies the Element in question can use it at all; and to the best of everypony's knowledge, there are only six known ponies who can use them, one per Element. The majority of citizens are upper-class unicorns.
Cloudsdale: This pegasus city, some 10,000 feet above sea level, manages the weather in Equestria. It is made almost entirely of clouds. It's also home to the Wonderbolts, the Equestrian air force. Any non-pegasi must have a "Cloud-walk" spell cast on them before travel. Rainbow Dash, the Element of Loyalty, is from here, but she currently lives in Ponyville.
Everfree Forest (BC): At the southern edge of Ponyville, the Everfree Forest contains various creatures, including dragons, hydras, timberwolves (wolves that are actually made out of timber) and cockatrices (sometimes called slither-hens by ponies). Fluttershy, the Element of Kindness, lives in a cottage on the forest's edge. The path next to her cottage leads to Zecora, a zebra herbalist known for always speaking in rhyme.
Ponyville: This town, located southwest of Canterlot, has an about-equal mix of earth ponies, pegasi and unicorns. The current bearers of the Elements of Harmony are usually found around here. Pinkie Pie, the Element of Laughter, works for the Cake family at the Sugarcube Corner Bakery. Rarity, the Element of Generosity lives and works at the Carousel Boutique, making dresses. Twilight Sparkle, the Element of Magic lives in the Golden Oak Library.
Sweet Apple Acres: This farm on the outskirts of Ponyville is also Equestria's premier apple orchard, famous for its rainbow-coloured Zap Apples. It's currently run by Applejack, the Element of Honesty.