You can have most of the products under your own brand name even with a brand new design! The minimal order requirement varies according to the different products.
ODM shorts for original design manufacturer, who designs according to your requirements and make your products based on the design. OEM shorts for original equipment manufacturer, who only make your products and you need to provide the product design for them at first. In some industry like the smartphone, the border between OEM and ODM are disappeared and most manufacturers provide design services as well as manufacturing.