Tape Looms and Woven Tapes
Tape looms of various sorts were used from ancient times through the early 19th century to make straps, tapes (a period term for what we'd call ribbon) and other narrow woven goods. Most traditional tape looms weave a warp-faced tape and use a rigid heddle to change the warp shed. The tapes woven on these looms ranged from very narrow tapes used for lacing to wider tapes used for straps and binding, as well as fancier edgings such as fringe.

Woven tape was used for items such as garters, apron strings, belts, ties for pillow bolsters, straps for powder horns or hunting pouches, carpet binding, trim for women's pockets, ties for seed or grain bags, loops for hanging up towels -- the list goes on and on. It was generally woven at home, but sometimes was available for sale from rural stores (Gehret 239) or from beggars (OED).

A Word About Inkle Looms
The term "inkle", according to the Oxford English Dictionary, means"a kind of linen tape, much used for various purposes"; the derivation is uncertain, but the word may come from the Dutch enckel, meaning 'single', which probably referred to the narrowness of the tapes. Other sources attribute the word to Scots or French sources, but since weavers and wares from the Low Countries wound up in various locations throughout Europe, it's not inconceivable that the word travelled as well. So, narrow woven tape is properly referred to as "inkle" in the 18th century. That does not mean that the modern loom known as an "inkle loom" is also correct.

The modern inkle loom was designed in 1924 by the Industrial Arts Cooperative Service (later known as Artcraft). I've heard people at reenacting events claim an older provenance for this style of loom, but these claims are spurious. That being said, modern inkle looms weave tape in the same way that older-style rigid-heddle looms do, except that they use string heddles to change the shed, making them easier to manufacture than a rigid heddle loom. The tape woven on inkle looms is not necessarily "wrong" for 18th century reenacting; the loom itself, however, is not appropriate to the period and should not be brought to events.

Pictures of Period Tape Looms
Some pictures from the Mercer Museum, dates unknown (probably 18th through early 19th centuries); unfortunately, these were displayed behind glass, so there's a bit of glare from the flash:

Tape looms from the Landis Valley Farm Museum, a museum that documents Pennsylvania German culture and history:

Also see: tape loom in collection of Pocumtuck Valley Memorial Association, Deerfield MA

More info and pics at Tapelooms.com

Recommended Reading

Neher, Evelyn, Inkle -- has pictures of tape and tape looms from the middle ages through modern times, as well as pictures and descriptions of people using the looms

Gehret, Ellen J., Rural Pennsylvania Clothing -- includes 6 pages of pictures and descriptions of 18th / early 19th century linen tapes

Weidert, Bonnie R., Tape Looms Past and Present -- nice survey of the subject, includes pictures of period looms, plans for making a loom, directions for setting it up, and weaving drafts

Weaver, Susan Faulkner: Handwoven Tape: Understanding and Weaving Early American and Contemporary Tape