Before the twentieth century, currency often was only loosely connected to the government of a country. In centuries past, money was often minted from gold and it contained the image of the king or other ruler. The coin contained a specific amount of gold (or another precious metal, such as silver). The main purpose of the king's image was authentication and verification. Essentially, it was saying: Someone trusted by the King has weighed this coin and has testified that it contains exactly one ounce of gold (or one half, one quarter, one tenth, etc). Such coins were commonly accepted not just within the king's jurisdiction but anywhere in the world. And they continued to be accepted long after the king's death.
Even today, governments issue golden coins verifying the exact gold contents of the coin. Since few kings are left, other national symbols appear on these coins: The American Eagle, the Australian Kangaroo, the Canadian Maple Leaf, the Austrian Vienna Philharmonic, are just a few examples of such coins being still issued.
Later, someone, somewhere, came up with the idea of not carrying gold on you all the time. Paper money was born. This money was essentially a certificate of deposit, a bearer note, still backed by gold (or silver, etc).
The situation changed in the 20th Century. Stalin and Hitler both abandonded the gold standard and introduced fiat currency (fiat is Latin for "be it so" implying that the money has no intrinsic value but is a representation of value because a dictator said so). Other countries followed. The last country to go 100% fiat was the United States in 1971 at the order of Richard Nixon.
The fiat currency is no longer backed by gold but by "the full faith and credit of US government" in the United States and similar ideas in other countries (for example, in the former Communist countries it was backed by the labor of the working class). Because of that, we have grown accustomed to see all money as a "gift" from the government (even the euro, though not a national currency, exists only by agreement of governmental entities).
But private currency is not only possible, it does exist and has always existed. Whenever a group of people (individual or corporate) agrees to base their exchange on something other than government issued currency, they are creating a private currency. This happens more than we realize, though usually it is a one-time thing ("I'll trade you this for that").
The emergence of Internet commerce has seen a rise in private currency. There are several challenges to e-commerce as far as currency is concerned. For one, it is not practical to just hand over paper money. As a buyer, you cannot insert a dollar bill inside your computer and have it transferred to the seller's computer (who may not be anywhere near his computer at the time of the transaction anyway). Nor can you do that with a check.
You can enter a credit card number, but that brings about a whole new slew of challenges: Not everyone has a credit card. Not everyone is willing to entrust his credit card number to a huge network of computers (never mind the fact that if you use a credit card at a physical store, the merchant enters its number to a network of computers). For years, banks were reluctant to let merchants accept credit card payments over the Internet. And, of course, the seller is hit by some serious fees for accepting credit cards and is forced to raise the price accordingly. Not to mention chargebacks, credit card fraud and similar problems.
Yet another challenge is the fact that e-commerce is global but fiat currency is not. Why should an Australian buyer pay a Japanese seller in US dollars? Why should they have "full faith" in US government? Or any other foreign government (we will leave the faith in one's own government out of this)?
As a proposed solution to the e-commerce challenge, various private currencies have been introduced. Naturally, if an international buyer does not want to rely on the full faith and credit of a foreign government, he certainly does not want to rely on the full faith and credit of an individual or a corporation. Because of that, a typical Internet private currency is fully backed by some other value. The most popular ones seem to be backed by the age-old means of global exchange: gold.
To the best of my knowledge, the oldest gold-backed Internet currency is e-gold, which is 100% backed by gold bullion. However, while gold bullion typically comes in 400 oz bars, e-gold allows anyone to own small fractions of these bars. It has a sophisticated shopping cart interface, so Internet buyers can pay Internet sellers with any amount of e-gold. While other gold-backed private e-currencies have risen, e-gold seems to remain the most popular and widely accepted. This, in my opinion, is because their shopping cart interface is superior to anything the rest of them have to offer (at least as of this writing - 2002-04-14). I wrote a description of this interface but because it uses too many illustrations, instead of noding it here, I posted it on www.goldcowboy.com.