Furoshiki come from Japan originally, and have been used for thousands of years. To begin with they were used by nobility, to wrap and store precious items, and later were often used at bath houses, marked with a family or household emblem and used to keep a bather's belongings together. At this point the word furoshiki was applied to them, from furo meaning bath, and shiki meaning cloth or spread.
Over time, the use of furoshiki spread to other tiers of society, with traders using them to transport their goods, and travellers carrying them far and wide. With such a long and rich history, it's not surprising that there are a great many techniques that can be employed, and there can be enormous significance behind the cloth chosen, the way it used, and the manner in which it is presented.
Nowadays, furoshiki are becoming popular as an alternative to disposable wrapping papers and gift bags. As in previous eras, the cloth itself can tell a story about the gift you're giving, or the person you're giving it to, and can even be a part of the gift itself.
Image: Example of furoshiki usage during Muromachi period (Image from: http://tutumimasu.exblog.jp/11015773/)