Bachelor's degree
Until the 19th century, a bachelor's degree represented the first degree in a particular faculty, with Arts representing undergraduate study, thus the Bachelor of Civil Law (BCL) at Oxford and the Bachelor of Laws (LLB) at Cambridge, for example, were postgraduate degrees. Vestiges of this system still remain in the ancient universities, with Oxford and Cambridge awarding BAs for undergraduate degrees in both arts and sciences (although both award undergraduate BTh degrees through associated theological colleges, and Oxford awards BFA degrees in addition to the BA) and defining other bachelor's degrees (e.g. BPhil, BCL) as postgraduate awards equivalent to master's degrees, although many postgraduate bachelor's degrees have now been replaced by equivalent master's degrees (e.g. LLM for the LLB at Cambridge and MSc for the BSc at Oxford). The same historical usage of indicating an undergraduate degree by it being in the faculty of arts rather than being a bachelor's degree gives rise to the Oxbridge MA and the Scottish MA).
Diploma of Higher Education
The Diploma certifies that a student has achieved a minimum standard for 2nd Year courses in science and liberal arts. If a student is undertaking a full Bachelor of Arts, a Diploma of Higher Education marks two-thirds of their undergraduate degree. Thus allowing the British Diploma to rank ahead of the American Associate of Arts (which in American parlance is a two-year undergraduate, or half of a BA/BS). That being said, American universities generally consider UK Diplomas equivalent to a college level Associate of Arts even though a British student typically holds more credit hours (and therefore more study time) in the arts, humanities and/or sciences.