Ideally you should use professional packing cartons, at least double wall construction. Boxes should be an appropriate size for the weight of your items. e.g.
Small boxes - great for books, canned food or other heavy items
Medium boxes - for lightweight items like clothes or toys
Large boxes - for oversize, light items like bedding or lamps
Old fruit and vegetable boxes from supermarkets should not be used - they tend not to have lids and can often be made of thin, single wall card which can tear, split or be crushed when moving your items causing risk of damage.
It's OK to use plastic crates providing they have lids and can be easily stacked.