The short answer
Cabinet is the widest term. It describes a storage case that commonly uses doors to enclose shelves, although drawers can be part of it too. Cabinets range from narrow cupboards to tall display pieces.
Chest has two common meanings: a low box with a lid, sometimes called a coffer, or a chest of drawers. Sideboard is more specific: a relatively low, wide storage piece designed to keep useful objects below while leaving a practical surface above.
Choose a chest for divided or deep storage
A chest of drawers gives separate, visible layers for clothing, linen or smaller objects. A lift-top chest offers one deep space, but anything placed on its lid must move before it can be opened.
Check the internal depth of the drawers or box rather than relying on the exterior. Thick timber, raised bases and drawer construction can reduce the space that is actually usable.
Choose a cabinet for shelves and flexible access
A cabinet lets you open a door and see several shelves together. It works well for objects of different heights, especially when the shelves can be moved or removed.
Measure the clear shelf width, depth and height, then allow room for the doors to swing. Glass-fronted cabinets display what they hold; solid doors keep the room visually quieter.
Choose a sideboard for storage and surface space
A sideboard combines cupboards or drawers with a broad top that can hold lamps, serving pieces, art or everyday objects. Its lower profile can sit beneath a window or leave wall space for a mirror or picture.
Sideboard, buffet and credenza are sometimes used for overlapping forms. The name can suggest history or intended room, but the doors, drawers, dimensions and construction tell you how the piece will work now.
Compare the usable inside
Before choosing any form, measure what must fit inside and record the clear internal dimensions. Operate every door and drawer, and check hinges, runners, locks, keys and handles while the piece stands level.
Allow space in the room for door swing, drawer extension and access to the top. Then measure the delivery route and confirm which shelves, doors, bases or tops can be removed safely.