同意 is the general verb for 'to agree' — you use it when you share someone's opinion or consent to a plan. It's less formal than 赞成 (zàn chéng, 'to endorse, to approve of') and much less official than 批准 (pī zhǔn, 'to approve/ratify' by an authority). 同意 can take an object (同意你的意见) and is negated with 不.