Belong means to be liked and welcomed by a person or group or that a thing should be placed in a certain area.
Ex: Mathew doesn't feel like he belongs with the kids at school. He likes books and video games, but they all like sports and being active, so they tease him and call him a nerd.
Ex: When I come back to this room, your toys better all be put away where they belong, or you're grounded, young lady!
If something belongs to you, it is your property, and you are the owner of it. You can also belong to a club or organization if you are a member.
Ex: Does anyone know who this lost puppy belongs to? I saw it wandering around the neighborhood yesterday, and today it came to our door begging for food.
Ex: Lolita belongs to the Honors Society, and she is selling cakes and pies to raise money for the school. Do you want to buy one?
When referring to buildings, businesses, money, or other commodities, to change hands means to transfer something to another owner.
Ex: The country hopes that the recession is finally coming to an end. A good sign of economic growth is that today at the stock market, a record number of stocks changed hands.
English has many expressions that include the word hands. Another common one is to take something into one's own hands, which means that someone will deal with something on his own.
Ex: The legal system exists for a reason; to restore balance to a society. Citizens should not take justice into their own hands, because actions taken in revenge only bring more violence.
Ex: Dennis was tired of waiting for his supervisor to recommend him for a promotion, so he took his fate into his own hands and went directly to the company's owner to ask for the higher position.