Depending on the volume of compost you desire you can use a trash can with a lid as the box. Important thing to remember is movement of the material while composting, container is somewhat secondary.
The ideal compost box will have an open bottom with strong walls and possibly a removable covering. Depending on the area where you wish to build your compost box it can be open chicken wire and a few stakes or a wooden walled box with space between the slats for venting.
A simple compost box can be made from 2x4's and MDF. Choose your dimensions and build a bottomless box. You can choose to leave the top open or put a hinged lid on.