Ahri really depends on her mobility and Ryzes W (prison) counters that. Ryze can push Ahri out of lane with Q. He outscales her, in addition to being impossible to kill.
Ahri cant trade with Swain, because of Swains abilities are easier to hit and does more damage. Before shes lvl 6, you can easily trap her with your snare and she cant escape. When she gets her ult, she can decide if she wants to stay and fight (lose) or
Movespeed on your e allows you to easily dodge her skillshots. You have more lane sustain because you dont use mana and your stun allows you to run in do some damage then run back out.
Veigars AOE stun is easy to get ahri with and as of the June 17, 2012 patch Veigars stun should now work (correctly) even if Ahri tries to dash through it. He also does damage based on Opponents AP and has 2 target-able skills to hit her with despite her mobility.
Morgana is heavily dependent on landing her Q then Pool for damage, which Ahri can easily dodge. After dodging, Morgana doesnt have the mobility to escape counterattack. In addition, Morganas ultimate is easily escaped by Ahris.
Karthus depends on opponents coming near him so that its easier to hit Qs and E. Ahri can jump around and is hard to catch Karthus wont have the damage to trade with Ahri. Ahri also has stronger CC
Xerath has long range and until Ahri gets her ulti, she should not be able to win the lane. When she gets it, tho, she will be able to avoid his skills and get near him. Xerath does not have any escape mechanism or shield, and therefore relies heavily on
Lux has pretty long cooldowns, so in a trade or teamfight she will need to land those spells. Ahri, with her Spirit Rush (R) can dodge all 3 of Lux spells. Ahri just has to play it safe until level 6, when she has a lot more mobility and sustain than Lux
In lane ahri outranges everythng but his silence. just wait till he blows that than quickly combo hm then back off. Easy lane win. you can also use the movement from your ult to escape his ult.
Huge kill potential will ultimate, mobility and burst. Ahri can countergank quite well due to her ultimates. TF lacks escapes and tankyness, to stand a chance against her.
Karmas ability to shield Ahri as she is diving towards the enemy team or ult-shield her if she is already inside their lines plus Karmas speed buff are all very helpful for Ahri to fulfill her role (jump around safely inside the enemy teams lines causing havoc).
If ahri hits a champ with her charm it will not only make them walk out of their team toward her but also slow their movespeed making it incredibly easy for blitzcrank to land a rocket grab afterwards securing an easy kill for your team.
