How is this not a fiat currency?

Bitcoin is absolutely fiat. Nothing backs bitcoin beyond what people are willing to exchange for bitcoin right now. 

Gold has a long, long history of holding value, bitcoin does not. However, gold has a long history of being confiscated too. The Roman army did it for years to continually expand the Roman empire. More recently, in 1933 US Presidential Executive Order 6102 prohibited individuals from "hoarding" gold coins, bullion, and certificates in excess of $100. People were required to "deliver on or before May 1, 1933, to a Federal Reserve bank or a branch or agency thereof or to any member bank of the Federal Reserve System all gold coin, gold bullion, and gold certificates now owned by them or coming into their ownership on or before April 28, 1933." Today, they simply arrest you for suspected 'smuggling' or 'money laundering' at airports and border crossings if they catch you carrying a lot of gold.

People looking at Cyprus are seeing that central banks are willing to confiscate and/or withhold normal paper money too. What is safe?

Bitcoin is fiat, but this fiat may be advantageous. If the gov't or a thief holds a gun to your head, they can't confiscate all of your bitcoin private keys unless you decide to give them all up. Since bitcoin is semi-anonymous, they can't easily tell who owns private keys of high value accounts to determine who to hold a gun to in the first place. If you were to give them one of your private keys, will they know that you actually have additional private keys containing further bitcoins that they could have confiscated?

The government could outright ban the use of bitcoins, making bitcoins pretty much worthless. Unlike confiscation of gold, however, the government wouldn't be gaining anything of much value in this case. All they would gain is people using government issued fiat again. The government has to print extra fiat to steal from people in this case. Too much printing (aka stealing) will encourage people to be lazy and stop working so hard. Why work hard to earn excess paper money to put away in savings if the government will inflate away the value of paper money before you get around to spending it?

What if someone hacks the algorithm?

Everybody relies on strong cryptography nowadays. If someone were to "hack the algorithm," say goodbye to the Internet and traditional electronic banking/e-commerce as we know it too. Bitcoin is based on the same strong cryptography that everybody else is using.

A bigger problem is someone hacking the computer that you use to manage your bitcoins. There are a lot of trojans/viruses trying to grab people's bitcoins right off their computer. This is why it is important to set up encrypted, off-line, cold storage for private keys that you only use from a secure computer. Ideally, what we really need are secure off-line bitcoin devices to store private keys and securely sign transactions when you want to spend bitcoins. Until then, a secure Linux Live Bootable DVD is probably the best thing to use for now.
Bitcoin will never catch on as a currency because there just isn't enough of it.

Using that line of reasoning, maybe people should move their savings over to Japanese Yen? They are going to print plenty of it to go around. There will be more than enough Yen for it to "catch on."

Bitcoin is not being inflated to death, unlike most government fiat today. Bitcoin would probably crash and become worth much less if the central banks were to suddenly stop printing money and stop confiscating people's bank deposits. If they keep on outright stealing from the people, bitcoin might just continue to gain popularity.

Is Bitcoin a Pyramid Scheme?

The earliest adopters of bitcoin were able to obtain bitcoins by mining them very easily. Early on, the block reward was 50 BTC every 10 minutes, now 25 BTC every 10 minutes, and will continue to drop in half every 4 years. Furthermore, as more adopt bitcoin and take up mining, the block reward is divided among more people, with each getting far less than those early adopters who divided the block reward among a comparatively small number of people. Those at the top (earliest adopters) get to reap great rewards, while those who are last in the door have to work very hard to get anything at all. Early adopters are probably sitting on large sums of bitcoins, waiting for more and more people buying bitcoins to push the price up. Someday, when those early adopters decide to cash out, they will dump their bitcoins and make off with a lot of money. Will it be enough to drop the price people are willing to pay for bitcoins to zero? Will it cause people to permanently abandon using bitcoin as a currency? If yes, then bitcoin may well turn out to be just another pyramid scheme.

It could also be argued that early adopters were developers and testers that spent a lot of time building a secure alternative currency infrastructure, without any upfront compensation or promises of eventual compensation for their work. Perhaps their rewards will simply be the price we pay for the work they did to bring this invention into the world.

If government cannot print, tax and regulate Bitcoin, they will ban it.

The government can print, tax, and regulate bitcoin very easily, even better than with traditional paper money. When you receive paper money from someone, you are on your honor to report your earnings and pay income tax. The same is true for sales/use tax in this situation. Only electronic transactions can fully and automatically enforce taxes.

When you make transactions using bitcoin, the bitcoin client software adds a transaction fee for you (i.e., sales tax/VAT). Even if you hacked the client software to remove this fee requirement, bitcoin miners' software ultimately decides whether to accept your transaction into the official transaction blockchain. If you don't pay a high enough transaction fee, your transaction could fail to ever get accepted into the blockchain and thus, never goes through. Software could just as easily enforce an incoming bitcoin fee if it were written to do so, providing a source of income tax. Right now, every 10 minutes the bitcoin miners print themselves 25 bitcoins. These could easily be changed to anything and directed to any government, if they could convince over 50% of the bitcoin network to agree to it. The source code literally is the code of law in a world running on bitcoin.

Being a bitcoin miner is akin to voting and governing. When over 50% of the states voted for Obama instead of Romney, the government continued to print, tax, and regulate US dollars under loose money policies. $85 billion dollars are printed per month (that's about $19.4M every 10 minutes). Hopefully, people mining bitcoin continue enforcing promises of minimal transaction fees (taxes) and minimal money printing.