Deli containers are plastic cylinder shaped containers designed to hold food. They come in various sizes including but not limited to quart, pint, cup, and two quart.
Depertment of Health regulations will vary from state to state, but in general all food product in containers must include a label with what the product is and the date that it was made generally written in sharpie marker and taped to the side.
Quart containers are nice for components such as dressings, aiolis, chopped garnishes, and other station specific mise en place that a cook will go through a decent amount of during a service. Things that are made in bulk during prep time are generally portioned into multiple quart containers and labeled for quick access during the rush of service.
An advantage of quart and pint containers is that they are pourable, so if you need to fill a squeeze bottle, it's much easier to pour from a quart versus a two tubby or lexan container.
An important note about quart containers that cooks learn is that they map perfectly to a 9 pan. Likewise a pint fits nicely into a shallow 9 pan and a two tubby fills a 6 pan.
Quart containers are perhaps the most prevalent and can often be seen serving the dual role of holding water for cooks during service. Note that beverages must have a lid, be labeled, and not reside near other mise en place during service.
Two quart containers otherwise known as "tubbies" or a "two tubby" are good for mise that you move through quicker or that you need a larger quantity of - items such as soups and ragus.
Pint containers are nice for more specialized mise that are made by the line cooks for their station for service like salsa verde, gremolata, reduced demi glace based sauces, and other fine knife work components like chive mince, parsley chiffonad, shallot mince, etc.
Cup containers also called half pint containers are handy for even finer components or those which you wont use a great deal of. Thyme leaves for example.
A general principle of back of hose work called "consolidation" dictates that you should use the smallest container possible for component inside. For example, at the end of service, if there's a cup of dressing left in the quart container, the line cook responsivle should scrape the dressing into a half pint container and create a new label for it. Chefs refer to this as working tight. Lots of room in lowboy refrigerators and walk-ins can be wasted by empty container space so a common task for junior cooks is to "consolidate the walk-in".
Containers that are held at room temperature during service should have the proper department of health label and generally be discarded after four hours. DOH regulations will vary by state but in general the label should include the name of the item, the temperature it originated from, the date, the time it came out, and the time it is to be discarded. Eg: saffron aïoli, 9/27, 36 deg, 5:00pm - 9:00pm.