What are the advantages of Bitcoin, compared to fiat currency (like euros and dollars) or commodities (like gold and silver)?

• Bitcoin is not owned, maintained, administered, or controlled by any authority, bank, FED, mint, government, company, or institute.
• Nobody owns Bitcoin. Bitcoin is not a company, or a website, or a program, or a service provided by someone. Essentially, Bitcoin is a protocol, that is effectuated by millions of people all over the world (by means of any software or implementation of their choice).
• Bitcoin is global. It is not limited to a certain region, area or distance. Paying your neighbour is just as easy as sending money to the other side of the planet.
  
• Bitcoin is transparent, verifiable, yet anonymous at the same time. Anybody can see and verify any transaction that takes place. They just can't be linked to actual persons.
• Storing and securing bitcoins is easy and safe.
• There are no bank accounts involved, no registered users, and no databases. You don't need to register anything anywhere.
• Bitcoin cannot be shut down. It does not depend on any server, database, office, or provider. It only depends on 'the internet' and the millions of Bitcoin users throughout the world.
• Bitcoins can be backed up easily (and when using encryption, this is much safer than any physical vault).
• Transferring bitcoins is fast, easy, and very cheap (virtually free). Compare this to the ridiculous processing times and rates charged by banks and credit card companies.
• Bitcoins can be transferred directly from one person to another, without depending on intermediate services or payment processors (who may impose their own rules, restrictions and outrageous fees).
• Bitcoins cannot be counterfeited. Transactions cannot be faked. (oh really? why?)→ What constitutes Bitcoin is merely a global list of transactions (stored redundantly throughout the entire network) which have to meet certain cryptographic conditions. Even with all the computing power in the universe, it would require calculating for trillions of years to meet these conditions (thus faking a transaction or stealing someone's money), unless you possess the private key to an address (or 'wallet') containing the money you want to spend. It's not protected because some software is supposedly difficult to hack (there is nothing to hack: all software involved with Bitcoin is completely free and open source) but because of the underlying mathematical principles. You can cheat people, you can hack software, but you can't hack math.
  See also this illustration about the level of cryptographic security of Bitcoin, which is much, MUCH stronger than with any other financial protocol in existence.
• Bitcoins cannot be created out of thin air at the press of a button (like banks and governments tend to do with fiat money). The mining process through which new bitcoins are created, is a controlled, predictable, limited, steady process. It doesn't matter if more people start mining, or if their hardware gets faster. The same total amount of bitcoins will be created, no matter what.
• Transactions cannot be reversed (like credit card chargebacks, paypal disputes, or undoing wire transfers by banks). This makes Bitcoin a very attractive payment method for stores and sellers. You can trust any payment you receive, even if you don't trust the payer.
• No authority can freeze your account (actually, there is no account), seize your money, block transactions, or prevent you from paying whatever* to whoever you want.
  (* of course limited by the amount of money you have)
• The supply of bitcoins is limited and predetermined. Only 21 million bitcoins will ever exist, this number will be reached around the year 2140. The certainty of knowing how many bitcoins exist today or will exist at any point the future, makes it a very stable currency.
• Bitcoins can be easily granulated, i.e. divided into smaller units. So even though the total amount of bitcoins is limited, there is no shortage of units. Current clients support up to 8 decimals (one hundred millionth of a bitcoin) which is more than enough for the foreseeable future. This can be easily increased further when the need arises.