The Liberty Dollar is an attempt to create a private currency in the United States. Basically, individuals can decide to accept or reject this currency of their own free will. Don't want to accept Liberty Dollars? Then don't. Want to accept them? Then do. It's that simple. Liberty Dollars look nothing like US government issued currency and they don't say 'legal tender' on them, so using them is sort of like using a foreign currency -- so long as you trust the issuer (if you don't want to accept foreign currency, you don't have to, but if I offer you £50 for a cup of coffee, you'd be fucking moron to say 'no').
The idea behind the Liberty Dollar is that it's backed by real gold and silver, not just trust. The Federal Reserve Note that Americans spend so freely isn't backed by precious metals, it's backed by faith and it's known as a fiat currency. The Liberty Dollar is backed by gold and silver. Well, that's the theory, but more on that later. The folks behind the Liberty Dollar have horded a lot of gold and silver stashed and the idea is that the Liberty Dollar represents a share of that horde.
In fact, people get surprised when they find out this is legal. The US Constitution gives Congress the right to issue money, but doesn't give an exclusive right (there used to be many competing currencies in the US and that used to be a huge problem). However, people are happy with Federal Reserve Notes and generally stick with them. However, there's a bit of a rub in the current system. As many folks know, a government can try to pay bills by simply printing extra money. This causes a huge problem. The government quickly pays off bills but all of the extra money flooding into the system creates a huge amount of inflation and prices rise. Thus, many people want a gold and/or silver standard. You can't issue new money without gold or silver to back it up, so a gold standard helps to limit inflationary pressures.
(As a side note, one might assume it's possible for a fiat currency to be immune to inflationary pressure simple by doing the same thing as the gold standard and tying it in a one-to-one relationship to labor. Unfortunately, labor, unlike gold, can not be measured exactly).
So why the hell don't we have a gold standard? Well, as you might suspect, there's no such thing as a free lunch. The problem with money is that if it is not spent, it's worthless. Anyone selling a home which no one is buying is painfully aware of this problem. Money, as a proxy for barter, must move through the economic system (in economics, the rate of monetary flow is referred to as the "velocity" of money). With the Liberty Dollar, it must move through the economy for it to be useful and, in fact, some folks have started to use it. However, what's the paper worth? If it's based on a precious metal standard, it must be fully redeemable for that precious metal or it's nothing more than a faith-based currency.
So what? Well, I've no idea if the folks behind the Liberty Dollar will cash in your money for gold or silver and if they don't, their money isn't worth the paper it's printed on. If they will, this creates the problem that led most countries to abandon precious metal standards. You see, if you can redeem the money for the precious metal, and you can acquire enough of the metal, you can corner the market. All of a sudden, you have total and complete control over the money supply. If one nation decides to pull of this stunt and suddenly declare that they don't want to honor another nation's notes based on the gold standard, a serious economic crisis develops. You don't get inflation, but you do get groups who can work to dominate the money supply and countries find that they don't have control over their economy.
Ultimately the problem is that precious metal standards actually make the economy more complex. We can't all go around trading chickens for car repairs because we might not have chickens or the mechanic might not accept chickens. So what we do today is accept money as a proxy for labor because this is much more efficient than bartering. Thus, you agree that X amount of money is worth Y amount of labor. With a fiat money supply, that's all there is to it. However, with a precious metal standard, you agree that X amount of money is worth Y amount of the precious metal which is worth Z amount of labor. You've introduced a more complicated system and as anyone who's worked in computer systems can attest, the more complicated a system, the more open it is to being compromised. In this case, one can compromise the system by dominating the market for the precious metal in question (as China once did when Britain was on the silver standard) and force those who value the metal to switch to another precious commodity.
So the precious metal standards fell because they introduce an extra layer of indirection between money and the labor it represents by inserting something "valuable" between them but complicating the system. This allowed others to dominate the market by acquiring that valuable thing. Of course, if extra supplies of the "valuable" item become available, then all currency gets instantly devalued.
You may think that fiat money is bad, but it eliminates the extra "valuable" layer and says "this is worth Y amount of labor". It makes things simple. Yes, it's prone to inflationary pressure, but at least a government has control over that pressure. They have little control over what people do with the "valuable" extra layer of a gold standard.
So why do I bring up this boring stuff? Because it seems that the Federal Mint is apparently reversing itself and declaring the Liberty Dollar illegal. Frankly, I think this is stupid. If people want to use a different currency and they're willing to trade it, so what? Further, it rather looks like the Federal Mint is on shaky grounds. However, I doubt the Liberty Dollar folks have the money to fight the government, so the private currency market is likely to die.