I like your vocabulary questions, Lily. Are you writing a dictionary? :)
In standard British English:
As an item of furniture, 'wardrobe' is quite specific: it's used for hanging clothes in the bedroom. A typical wardrobe is maybe 2 metres high with a horizontal rail for hanging dresses, suits etc on hangers. It usually has a door or two, and may also have shelves and drawers underneath or at the side.
A 'cupboard' is a very general term for many different kinds of enclosed spaces for storing things. Cupboards generally have shelves. In kitchens, they contain food and cooking utensils; in bathrooms, they contain toiletries, and so on. At work, you might have cupboards for files and suchlike.
Both wardrobes and cupboards may be freestanding (moveable) or built-in (part of the fixtures and fittings of the room).
A cupboard can also refer to small storage room behind a door e.g. a broom cupboard where you keep brushes, mops, buckets and so on. In typical two-storey houses, there's often an 'understairs cupboard' for storage. In a work environment, such as a school, there might be a similar area known as a 'stationery cupboard' where paper and similar supplies are kept.