"Turns" refer to each time you and your opponent attack eachother.
Let's suppose your Leafkit is fighting a Pixicroak. Leafkit attacks, and then Pixicroak attacks. Those two "moves" count as one turn, meaning that poison can only wear off on its own only if you DO use your kin in battle, instead of letting it sit idly in your party.
Poisoned kins can still fight, but their poison status will drain their health bars by a few points each turn. After a certain number of turns (which may or may not be in the same battle), the poison wears off and your kin stops losing health each turn.
Also, instead of waiting for poison to wear off, you have the option to use an Antidote. They're dropped by wild kins as their battle loot, given as level-up rewards, or you can buy some really cheap in the marketplace.
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