How to play against Ahri:
Hard burst and CC and more or equal mobility.
Ahri is weak against:
Burst casters like LeBlanc can outright kill Ahri. In addition, LeBlanc has a slowroot to bring Ahris mobility down.
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.
Ahri is strong against:
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
Ahris ability to move around makes it easy to dodge anivias stun then you just burst combo anivia down to nothing.
Ahri synergizes with:
Rammus powerball pop-up and taunt easily sets up a charm.
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.
No special synergy here, except that any champion with an AoE stun is great with Ahri, since it allows her to deal damage while being safe from harm.