A bag is a container with handles that you carry around with you. You can put all sorts of things in a bag, depending on how big it is.
School children carry their books in a bag on their back. Women carry lipstick in small bags that they carry in their hands or over one shoulder - usually called handbags. The biggest bag is a suitcase that you pack a lot of your clothes in when you go on holidays.
A basket is a container that is woven, in a process called wickerwork. Baskets can be made from thin willow branches, bamboo, plastic or wire.
Boxes are something you store things in. When you buy an appliance from a shop it often comes in a box. Boxes can be made of simple cardboard, or can be very fancy wooden boxes.
A box that is big enough makes a great cubbyhouse. The box in the picture is big enough to fit a girl inside.
A briefcase is a rectangular bag that is most often used by professional workers. Briefcases are usually black and made of leather.