Vanille is a l'Cie, a person marked by a fal'Cie and given a Focus- a mission to complete. If a l'Cie fails to fulfill their Focus, they will turn into a Cie'th. If they do fulfill their Focus, they will turn to Crystal (sounds like a win-win situation, huh?). Vanille's particular flavor of l'Cie is a Pulse l'Cie, someone who is to do the bidding of the Pulse fal'Cie. Upon being turned into a l'Cie, they are granted many powers, including the ability to cast spells. For Vanille, this is especially true- she's not exactly the most physically strong character in-game, but her skills using magic are some of the best, particularly in healing, debuffing and casting elemental magic. She can cast all of the second level -ra spells, and Blizzaga and Firaga.
At her level in the Crystarium, here are the spells she would know; for simplicity's sake, I decided to limit it to her three main roles, though more would be available to her in the actual game.
Offensive magic, based around elemental spells. While Vanille can learn the basic and -ra level spells of each element, she can only learn the -ga level of fire and ice spells. Because of her canon point, she'll still be lacking spells of this level. Basic spells hit a single target, while -ra spells are Area-of-Effect spells that hit the enemies surrounding the target.
☀ Fire Fire, Fira
☀ Ice Blizzard, Blizzara
☀ Electricity Thunder, Thundara
☀ Water Water, Watera
☀ Wind Aero, Aerora
Healing magic, meant to restore HP to party members and revive from KO; can also remove certain negative status effects/debuffs, excepting Doom, which can only be cast by Eidolons.
☀ Cure Restores HP to a single party member.
☀ Cura Restores HP to a party member and the party members nearby (AoE spell).
☀ Curasa Restores HP to a single party member based on damage suffered. The lower the HP, the more is recovered.
☀ Curaja Restores HP to a party member and the party members nearby (AoE spell). The lower the HP, the more is recovered.
☀ Esuna Removes the last suffered debuff.
☀ Raise Revives character from KO, restores a small amount of HP.
Debuffing magic meant to enfeeble enemies by making them more susceptible to attacks by lowering their defenses and resistance, as well as slow their advance. The party's chance to inflict status effects is also increased. Vanille is a specialist in the Saboteur paradigm role, however she is unable to cast 'Death,' a spell exclusive to her, due to her canon point.
☀ Deprotect Decreases resistance to physical damage on a single target.
☀ Deprotega Decreases resistance to physical damage on a single target and surrounding foes (AoE spell).
☀ Deshell Decreases magical resistance on a single target.
☀ Deshallga Decreases magical resistance on a single target and surroundong foes (AoE spell).
☀ Poison Constantly damages the target's HP by draining a constant percentage of a single target's total health.
☀ Imperil Decreases a target's elemental weakness.
☀ Dispel Removes the most recently cast enhancing magic from an opponent.
Vanille also has the ability to summon an Eidolon, Hecatoncheir (which she adorably refers to as Hecaton). He is an Earth-element summon with multiple arms. Once summoned, he can help her in battle, and is the only Eidolon capable of using Quake, able to deal earth damage to a wide radius of enemies at once. The Eidolons can also go into Gestalt Mode, which allows the summoner to use them as a vehicle and ride them. Hecatoncheir transforms into a giant mech-type vehicle, which Vanille mounts. He has gun turrets and most of his attacks are ranged. His super move is Gaian Salvo, which is pretty much Hecatoncheir sticking it's guns into the ground and blowing it up. She does not have the ability to summon it from her canon point, but I decided to make note of it regardless.
Her weapon is the Binding Rod, which is, essentially... A fishing rod with antlers. She casts it out, it hooks onto enemies, and deals damage when she pulls it back. It can latch onto and deal damage to multiple enemies.