Unicorns
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Generation 1

Unicorns were one of the first species introduced and one of the three main species across the franchise. G1 has a lot of canonicities, so what Unicorns are like and what they can do varies by source.
On the toy backcards, Earth Ponies have magic like all other types of ponies. Whatever type of random magic you can think of, they can probably do it.
Only one pony is shown doing magic in the Rescue at Midnight Castle special and that is the Unicorn Twilight, who can teleport by saying "I Wish". This teleportation ability becomes standard among all Unicorns in MLP 'n Friends, where it is called "winking". Unicorns in MLP 'n Friends also have individual magic, such as Buttons' levitation or Gusty's wind magic. The first Unicorn in Dream Valley originated from the land of Argyte. She had four magic horseshoes that must be worn by her descendents in order to stay healthy, but most Unicorns do not seem to be closely related to her.
No Unicorns appear in MLP Tales, but one pony with both wings and a horn does.
Majesty in the UK G1 comics is able to do many different kinds of magic, but I don't know if that ability is exclusive to her or not. Any info is appreciated!
Generation 2

Only a few Unicorns appear in G2, and they are all toy-exclusive characters. While often associated with magic, their exact powers are unclear. They have the ability to grow magic butterfly-like wings, arguably making them alicorns of a sort. Unlike most generations, their horns are a different colour from their fur.
Generation 3

Unicorns in G3 live in the land of Unicornia. Every year they choose a "Rainbow Princess" (who has to be pink) to make the first rainbow of the season alongside a few other unicorns (who aren't pink) by touching their horns together. If the rainbow isn't created in time, then everything starts to lose its colour. Outside of this they don't seem to have any magical abilities, with the exception of Lily Lightly whose horn glows when she's happy. Sweetie Belle in G3.5 also gets a single shot of sparkles coming out of her horn, but it's never seen again.
Generation 4

Unicorns in G4 have a wide range of what they can do with their magic. While most Unicorns can only do basic spells like levitation and light, some Unicorns have additional magic related to their special talents, such as Rarity's gem-finding spell. Certain Unicorns have an extra affinity for magic, allowing them to do everything from teleportation to transforming ponies into other species to time travel. When a unicorn does magic, their horn lights up with an aura, the colour of which depends on the individual. Canterlot has a high population of Unicorns.
Vaguely Chinese-themed Unicorns with unique curved horns existed in the distant past, and one, Mistmane, still lives on in the present. Not much is known about them, aside from Mistmane they only appear in one episode, but some of them appear in the Gameloft game with the description "Imperial".
Generation 5

Unicorns were unable to cast spells for many generations, after the separation of the unity crystals caused a complete loss of magic. Once they were reunited, Unicorns regained their Float (levitation) magic, while pegasi regained Flight magic and earth ponies got Flora magic. Unicorns became superstition after losing their abilities, but started to loosen up once magic returned.
Since they have been without magic for so long, most Unicorns only know basic magic like levitation and light, but Unicorns start to relearn how to use their powers throughout the generation. The episode Winter Wishday introduces a vocal component to certain spells, something not seen before, allowing Unicorns to create frost or heat depending on what they say. Misty starts to learn a variety of spells not done by other Unicorns near the generation's end.
Most Unicorns live in the magical forest of Bridlewood, where they drink tea and listen to poetry. They are more nature-themed and hippie-ish than in past gens, and many have darker or more earthy coat colours. Their horns are different colours from their coats and feature elaborate swirl patterns.