You could think about the differences by picturing a debate:
1) Disagree - people involved in a debate may disagree with each other about the debate topic. This is a starting point - disagreement signals a difference of opinion, a different viewpoint, a belief that the other person is wrong.
2) Object - a person may object to what another person says, believing it not just be be mistaken but possibly false, untrue, whether through ignorance or deception. Objection is stronger than disagreement - when you disagree, you think the other person is wrong, when you object it may be that you think the other person is being dishonest or manipulative in trying to support their argument.
3) Complain - people complain when they feel personally aggrieved about something. In the example of a debate, a person might complain to the chair that someone on the other team is not respecting the rules agreed on for the debate, e.g talking over other people.
4) Criticize (or UK spelling criticise) - criticism can be directed at people or things, and it about how that person or thing is or does something. For example, in a debate one person might criticise another person by commenting on the weaknesses in their debating technique, or on how well prepared they came for the debate, or anything else.