If you have a general idea of what you want to make, these factories will help refine your idea and think through the different components that involve the making of your idea.
This stage involves gathering all the necessary materials needed to make your product. Some factories provide materials and assembly; sometimes you will need to buy materials separately.
A sample is a prototype of your idea. Sample-makers usually make the product by hand (hand-cutting, hand-assembling, etc.) before you get your tooling finalized. You want your sample to be perfect if you plan to do mass production of your product.
(Usually sample-makers charge more per product, but this depends on the factory’s policy.)
To produce products at a rapid rate and drive your cost down, you might have to invest in machinery that is customized to your design; that is "tooling". This machinery will vary depending on the product, but can range from plastic molds to custom branding stamps, and so on.
"Production run" means making a lot of one product at once. The factories listed under "production" can produce small and/or large quantities. The minimum order for a production run can vary from factory to factory (some have no minimum, some require a dozen, and some require up to 1,000 to be made at once)