A specific amount of resource, reserved for some specific purpose. Most of the time, the “resource” is money, and the allocation is to a WBS Element. However, the term “budget” can also be used in a technical sense, referring to some run-time resource relevant to the target system (e.g., “link budget” or “weight budget”).
The concept of a programmatic budget is completely distinct from that of a BoE, and the two often have no relationship at all beyond their totals…and sometimes not even that much.