Heihaku is a general term used for anything other than food and drink that is offered to the Shinto gods during a Shinto ceremony. In its broadest sense the term also includes even the food and drink offered to the gods.
In addition to 'heihaku,' offerings are also called 'mitegura' and 'heimotsu.'
The term 'haku' refers to cloth, and refers back to ancient times, when cloth was precious and thus became the primary item offered to the gods. In one article under Shinto ritual prayer in the "Engi shiki," the list of items described as heihaku included cloth fabric, clothes, weapons and armor, sacred wine, and food and drink offerings to the gods. Heihaku are offerings to the gods, but at the same time, they are thought to be yorishiro (objects in which the gods dwell), and so if these heihaku are treated like yorishiro such as poles with streamers or arms and armor, then they become heisoku (rope or paper offerings), gohei (a staff with streamers), and taima (paper offerings in Shinto), but such objects are also called heihaku.
According to the 'Jinja saishiki' (Shinto shrine rituals) established in 1875, it was decided that in addition to actual items made of cloth and so forth, kinhei (coins wrapped in paper) would also be considered heihaku. Today, for regular festivals of the shrines under the comprehensive purview of the nationwide Jinja Honcho (Shinto Shrine Association), the Jinja Honcho sends funds they call 'heihaku expenses' to the shrines.
Heihaku offered to the gods by order of the emperor are called hobei or hohei (presented offerings), and for that reason the messengers that the emperor sends to shrines are termed the "hoheishi" (messenger to present offerings; starting in the Meiji period, the name was changed to "heihaku gushinshi" or "messenger that gives heihaku as offerings"). The messengers sent by the Jinja Honcho to each of the various shrines to offer them heihaku are called "kenheishi" (messengers who offer heihaku).
A waka poem concerning heihaku
By SUGAWARA no Michizane: 'At the present time, the hei too do not join hands, at Tamukeyama Hachiman-gu Shrine, a brocade of crimson leaves, here along with the gods' (in the Kokin wakashu)