Allergies will cause HP Drain status while you remain near the offending object, usually plants or metal. Hater will lower your effectiveness around that individual or type of unit(save Imp Hater, which is a boost when near Imperial forces). Phobias will also lower your effectiveness.

Panicky will lower your attack power when facing enemy interception fire. Fatalist will lower your defense when low on health. The Child of Nature potential lowers effectiveness when standing on paved surfaces. Neat Freak does the same when standing on dirt.

Some specific soldiers have their own as well: Nancy's Klutz(you lose your grenade), Freesia's Under Pressure(don't use her for your final CP), Claudia's Unlucky(can't do critical damage on that attack), Suzy's Humanitarian(refuses to attack an enemy), Herbert's Moody(refuses to take any action, save for restocking ammo and repairing sandbags), Karl's Headlit Deer(lower defenses), Homer's Frail Body(triggers when he uses more than half his AP).

Remember that all potentials have a chance to activate. So you won't always trigger them.