Here's the simplest explanation: A cupboard is typically found in a kitchen and holds dishes, food, etc. The term may be a bit outdated in the U.S. but is used more often in the U.K., Canada, etc. A cabinet is used more often in the U.S. instead of the term cupboard (e.g., kitchen cabinets). But generally speaking, a cabinet is also found outside the kitchen and is a place to store things like garage tools or clothes or cleaning supplies. A closet is specifically for clothes and shoes. In the U.S., it's often a small room off a bedroom. Here in Europe, it's rare to find a house or apartment with closets; instead we use stand-alone armoires that are pieces of furniture. One can also use cabinets for the same purpose. Hope this helps!

In the UK we don't usually use the word closet. A cupboard is usually somewhere to put items like plates, clothes, food or things that you do not wish to be on show. The door is usually solid. It can stand on the floor or be attached to a wall. A cabinet is where you would put things like decorative plates, or other items that you are happy to have on show. The doors are usually made so you can see the things inside. It can be freestanding or be attached to a wall. There are also medicine cabinets that you have in a bathroom where the items can be stored away. In an office there is also a filing cabinet where files are kept.

In the US, closets aren’t only for clothes and shoes, though that would be the default. You can have linen closets, supply closets, storage closets, utility closets etc. Basically it’s a place to store things behind a door that goes from the floor to the ceiling or high enough to walk into. A closet in the kitchen to store food would be called a “pantry” not a “closet” though.