When creating a product, you have the ability to apply specific settings which control the visibility of the product both in the search results and what is displayed on the product itself. Here's a detailed list of how these settings work:
Make this product visible to the public: By checking this box it will make the product visible to the public (however it doesn't affect if it can be purchased or not). If you uncheck this box, then it will not be seen on your web site.
Limited quantity available, once it sells remove it from the site: Once the stock goes to 0, we will set the product has hidden (essentially checking the option above).
Allow backordering of this product if you don't have it in stock: When your stock is at 0, we will allow people to purchase this product, but they will be informed that it's on backorder.
This product is built on demand when ordered: When selecting this option, you will be prompted to enter the lead time. This would then be displayed to the person viewing the product, so they are aware their order will be delayed.
Flag this Product: you can set a flag that will adjust the placement of the product on your site. Keep in mind it will always show up in your search results/categories.
Featured Product: shows up on the feature product category page
Frontpage Special: shows up on the standard commerce home page
Discontinued Product: displays a flag to the consumer that the product is discontinued
Special Order: displays a flag to the consumer that the product is a special order
In Stock: the product is currently available to purchase and we will display the quantity available to the purchaser (if there are more than 25 available it will state 25+)
On Order: the product is not in stock (or could be) and there are more on order. The purchaser will see that it is on order and when the expected date is.
Please note that several of these settings may override others. For example if the product is hidden, it won't show up as a featured product.
The ONLY time a product would not be displayed in your product catalog is if it's set to hidden by unchecking "make this product visible to the public". Otherwise people will see it in your product catalog; however they do have an option to only show products that are available for purchase.