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)