"You're reckless, little one. You never would have made it as Obi-Wan's Padawan... but you might make it as mine."
- Anakin to Ahsoka on Christophsis
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Anakin Skywalker, the future Darth Vader, was believed by some to be the Chosen One of Jedi prophecy. Anakin spent his childhood with his mother on Tatooine, before his superhuman abilities and abnormally high midichlorian count caught the attention of Jedi Master Qui-Gon Jinn. Anakin's heroics in the Battle of Naboo led him to be inducted into the Order as the apprentice of Obi-Wan Kenobi, at far later an age than was generally allowed. His bold, rebellious nature and extraordinary abilities earned him a great deal of fame and notoriety at the Temple. This rebellious nature was part of what led him to, in defiance of the Jedi code, secretly marry his childhood love Padmé Amidala at the start of the Clone Wars. The brutal death of Anakin's mother intensified his deeply-rooted fear of losing those he loved, including Obi-Wan, Padmé, and eventually Ahsoka.
Anakin was Knighted shortly into the Clone Wars, his and Obi-Wan's success in many campaigns earning them fame as heroes throughout the Republic. At the Battle of Christophsis Anakin was surprised to learn the Council had assigned him a Padawan - Ahsoka Tano. Initially indignant and resistant to the idea of training an apprentice, Anakin soon grew very fond of his new charge, who in many ways was just as reckless as he.
Anakin, while protective of Ahsoka, also considered it important to show trust in Ahsoka's abilities - a confidence he felt he rarely got as Obi-Wan's student. Anakin has also taught Ahsoka to, above all else, trust her own instincts and feelings to guide her... and that while one should follow the Code, there are loopholes in it, too. There are crucial lessons Anakin remains unable to teach his apprentice, however, such as how to be able to let someone you care about die if a mission demands it.
Unbeknownst to Anakin or Ahsoka, there was a hidden reason behind her assignment to Anakin: the Council hoped that this would teach Anakin to be able to "let go" of his attachments to those he cared about. Unsurprisingly, this has turned out to be unsuccessful; as time goes on, the two's bond has strengthened. Anakin is unwilling to consider the possibility his apprentice could die, and has gone to great - and sometimes brutal - lengths to preserve her life.
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Ahsoka Tano is a Togruta from the planet Shili. She was discovered at a very young age by Jedi Master Plo Koon, who brought her to the Jedi Temple and who Ahsoka shared a fond relationship with. At age fourteen, shortly after the start of the Clone Wars, Ahsoka found herself apprenticed to the brash young Knight Anakin Skywalker. Though Ahsoka was eager to impress her new, famous Master, Anakin was initially reticent to take on an apprentice. After their first campaign together, he softened towards her, the two establishing a more formal rapport, though often teasing one another fondly.
Ahsoka, unlike Anakin, was brought up in the Temple, and often tried to ensure her Master's reckless streak didn't take them too far out of the bounds of the Jedi Code. However, Ahsoka's own reckless streak has increased over the course of her apprenticeship, her personality mirroring her master more and more. At times she has difficulty following orders - even Anakin's - is prone to occasional flashes of temper, and has shown some signs of inheriting Anakin's own tendency for deep emotional attachment. She has shown awareness that her Master has philosophies other Jedi would consider "radical," and it's clear she's concerned about the toll the war takes on him.
Ahsoka has shown herself to be a talented apprentice, capable enough that Anakin admits he can forget just how young she actually is. She is talented enough to fight alongside her master in many battles, even occasionally able to hold her own - albeit briefly - against Dark Side adepts like Asajj Ventress and the droid general Grievous. She has also successfully strategized several key military campaigns, such as helping break the Separatist blockade over the planet Ryloth.