Persuader is a social skill used not only by idle dwarves at taverns and meeting halls, but also by the fortress broker when attempting a particularly tight-fisted trade.
This skill can be trained by letting dwarves idle and chatter amongst themselves. Dwarves that are fairly unassertive will never gain experience in this skill.
In Adventure mode, this skill (along with judge of intent) is trained any time you state your values in conversation and, having encountered disagreement, press the argument. Note that it is only raised when the other party disagrees with your belief. At a sufficiently high experience level, others (aside from particularly close-minded NPCs) will be swayed by your initial statement of values alone - so powerful is this ability that elves can be convinced of the worthlessness of nature, and goblins of the value of harmony. If you're looking to raise this skill quickly, find an NPC who, in idle chatter, says "I'm not going to change my mind." Once found, you can convince them of one value, only to convince them of the opposite value once they've finally changed their mind, and repeat.