A 'border' is an area of ground where you have flowering plants, shrubs even trees it it is big enough. Normally used to describe something that is running along the side or back of the garden i.e. it has a backdrop of a hedge/fence or something else.
'Bedding' refers to annual flowers which are planted out after all risk of frost is past.
a 'bed' in the garden is another area for flowers etc. but stands alone. In other words it is surrounded by grass, gravel, paths.

Beds and borders can range from very small to very large.

Just a side note: beds that are surrounded by lawns or paving are sometimes called "island beds" to distinguish them from "border beds", especially in older gardening books. Modern usage is considerably more elastic and blurry.