PoE, PoE+, PoE++ …what is the standard behind the name?

PoE refers to the common name, whilst 802.xx refers to the norm defining the standards:

• PoE (Type 1) = 802.3af
• PoE+ (Type 2) = 802.3at
• PoE++ (Type 3 & 4) = 802,3.bt