Weakened is a status effect that temporarily reduces the afflicted creature's strength. Weakness lasts for 300 turns. If an already weakened creature is weakened again, the strength penalty will increase by 1 and the timer will reset.
For you, the impact of being weakened depends on the strength requirements of your current equipment. If your effective strength drops below the strength requirement of your armor or weapon, your defensive or offensive capabilities will suffer severely. Consider equipping a lighter item until the effect wears off. If your strength is still above the requirements of your equipment, then the impact is neglegible. You can instantly cure weakness by drinking a potion of strength or potion of life.
When creatures are weakened, they suffer penalties for each point of weakness as if their base strength was only just enough to use their built-in weapon and armor. Because of this, having an ally that can inflict weakness can quickly render a powerful foe mostly harmless.
You can cure weakness from an ally by empowering it.