1. record or make note of something, express a commitment to something
put my name down for something
Put down is a phrasal a verb with a number of different meanings. To put an object down means to put something or someone on a surface or at a lower level to where it currently is. If you put down a person, it means to be critical of someone in a public or derogatory way. If you kill an animal by using a lethal injection in order to make their death more humane you can say you put down the animal. Sometimes if you are talking about putting something down, it can mean to write something or to put something down on paper in order to preserve a record. If you put money down on something it usually refers to paying a deposit or to pay a fraction of the cost of something or to put money towards a specific purpose. When talking about transport to put down someone can mean to stop to allow someone to exit a vehicle. When you are talking about a plane landing you can also use the phrase a verb put down. If you put a young child or baby to bed you can say you put them down for the night. To formally ask a committee or department or official organisation to consider something you might need to put down a motion or put down a suggestion.
2. criticize or insult yourself or others in a harsh way
put someone/yourself down
The phrasal a verb put down or to put down someone refers to criticising someone or someones work in a derogatory, negative or embarrassing way.