User can create, shape and manipulate electricity, a form of energy resulting from the movement of charged particles (such as electrons or protons), allowing control over electric fields, all charge carriers (Ions, Electrons, Protons, and Positrons), and electronics. One could eventually grow to manipulate the weather and create thunderstorms.
Distance, force, precision, etc. depend upon the knowledge, skill, and strength of the user and their power's natural limits.
Electricity needs a conductor like metal or water to move through, therefore electricity can neither exist in nor move through a vacuum (such as outer space) and may be insulated by non-conductive matter, such as air and rubber.
May be unable to create electricity, being limited to manipulating only from already existing sources.
Electricity may be redirected by certain materials.
A strong enough magnetic force/source may be beneficial or a hindrance.
May be limited only to the path of least resistance.
May become useless if confronted with electrical resistant material, such as rubber, silicate, etc.
May have to find electrical sources of power in order to recharge.
In universes such as The Avatar series, electricity manipulation requires the user to be mentally stable with the complete absence of emotions and peace of mind. Physically, it requires separating the energies of yin and yang, which in our universe can be interpreted as positive and negative electric potential energy respectively. When the yin (positive chi energy) and yang (negative chi energy) collide together to become whole again, lightning is created. Essentially, it's similar to matter and anti-matter collision. In which when they collide, they release pure energy in the form of protons and electrons (photons are nothing but light, not electricity). Supposedly in the Avatar Universe, the combination of yin and yang release the pure energy in the form of electricity.
Electrokinesis is also the term used in Electrohydrodynamics to refer to the movement of a fluid or particle by an electric field in a fluid with a net mobile charge. To prevent possible confusion, one could use the term Fulgurkinesis to refer to someone with the power to manipulate lightning. However, the term Electrokinesis to refer to fictional users of the power is commonly used enough that it's unlikely to be a major source of confusion.
Because electricity can get extremely hot when lightning passes through the air, lightning can heat the air it passes through to 50,000 degrees Fahrenheit (5 times hotter than the surface of the sun).
The Black Garnet (She-Ra and the Princesses of Power)
Ambrosia of Indra (Shinju no Nectar)
Tezuka Jun (Alive: The Final Evolution) can manipulate electricity.
Mikoto Misaka (Toaru Kagaku no Railgun) imbuing electricity into a coin to shoot a Railgun...
...and blasts a metal structure in half with two coins.
Shundan Kimi (Toaru Kagaku no Railgun) is able to produce and use black electricity from her left hand.
The Thunder (CardCaptor Sakura)
Hitomi (Code: Breaker) controlling electricity.
Kanon Ozu (Coppelion) had an eel gene inserted into her, allowing her to generate and control electricity.
Skin Bolic (D.Gray-man) can become a behemoth of lightning.
Akira Otoishi's Stand, Chili Pepper/Red Hot Chilli Pepper (JoJo's Bizarre Adventure Part IV: Diamond is Unbreakable)
Fu Chi'en (Tenjho Tenge) can draw electricity from the earth and utilize it as he wants, so long as his feet are rooted to the earth.
Lambo's Thunder Set (Katekyo Hitman Reborn!)
Zatch (Zatch Bell!) using his signature spell, Zakeru.
Zeno (Zatch Bell!) can generate and project electric blasts more powerful than his brother, Zatch.
As an Electric-type Pokémon, Pikachu (Pokémon) is able to attack foes with its electric charge
Zekrom (Pokémon) manipulating lightning.
Ainz Ooal Gown (Overlord) using Dragon Lightning.
The Third Raikage (Naruto) had mastery over Lightning Release chakra, creating the powerful Lightning Release Armour technique.
Unruly Ay/A, the Fourth Raikage (Naruto) has great mastery over Lightning Release chakra.
Kakashi Hatake (Naruto) can convert his Chakra into electricity and generate it to the point where it becomes visible to create his original Lightning Cutter/Raikiri.
With mastery over his Nature Affinity of Lightning/Raiton, Sasuke Uchiha (Naruto) is famous for channelling his Chakra into electrical energy, forming his powerful Chidori...
...and through training he can imbued his Snake Blade with lightning energy to enhance the sword's cutting ability...
...and the pinnacle of his Lightning Release is to use it to manipulate natural lightning via his Kirin, which can destroy an entire mountain.
Omoi (Naruto) using Lightning Release: Thunderbolt.
As the Lightning Dragon Slayer, Laxus Dreyar (Fairy Tail) a powerful user of Lightning Magic.
Killua Zoldyck (Hunter x Hunter) demonstrating his Electric Nen...
...which is utilized through his various electricity-based techniques like his Godspeed.
Spirit of Thunder (Shaman King)
Gilthunder (Nanatsu no Taizai) can use his ability Thunderbolt to generate and manipulate lightning.
Thanks to her lightning-type Mana Conversion Affinity, Fate Testarossa (Magical Girl Lyrical Nanoha) is able to unleash a wide variety of electrical-based attacks and form her electricity into powerful scythes and swords.
Cure Peace (Smile Precure) can discharge large amounts of electricity. However...
...her powers go out of control whenever she cries. As a result, Cure Peace would (unintentionally) zap the opponent that is scaring her.
With the Devil Fruit powers of the Rumble-Rumble Fruit/Goro Goro no Mi, Enel (One Piece) can generate electrical charges up to 200,000,000 volts.
With her Clima-Tact, Nami (One Piece) can manipulate electricity for a variety of uses.
Charlotte Linlin/Big Mom (One Piece) controls a living thundercloud called Zeus...
...allows her to manipulate lightning for deadly combat.
By drinking the ambrosia of Indra, Kai Watari (Shinju no Nectar) gain the lightning powers of a Thunder God.
Sailor Jupiter (Sailor Moon)
Sinbad (Magi Series) uses lightning powers given to him by the Djinn Baal.
Trish (Devil May Cry) can manipulate electricity to project electrical beams, teleport, and amplify her physical attacks.
The Blitz (Devil May Cry 4) is an elite demon sired by Mundus. It is able to manipulate the electricity around its body to projects beams of electricity, teleport, and electrically-amplify its physical attacks.
Lightning Dinosaurs/Ceratopsids (Dinosaur King)
Razor, the Lightning Remnant (DoTA 2)
Lighting mostly has thunder magic. (Final Fantasy XIII/Dissidia)
Kratos (God of War) wielding Zeus' lightning bolt.
Kratos (God of War) unleashing Poseidon's Rage.
Elektricitie (Gravity Rush 2) lives up to her name and reputation.