Bitcoins aren't like gold. No one can dig them out of the ground. They're not like paper money, either. No central authority prints hard-to-counterfeit, tangible paper bills for circulation. Instead, bitcoins depend entirely on a decentralized computer network and some amazing feats of cryptography.
For starters, understand that the entire Bitcoin system runs on P2P (peer-to-peer) network. This P2P architecture is similar to file-sharing networks like the ones that allow people to freely distribute data of all kinds, including copyrighted music, movies and more. It's a resilient system.
In other words, there isn't a central computer hub running all of the bitcoin-related processes. Instead, each Bitcoin user's computer is part of the network, collectively sharing the computing burden of generating bitcoins and logging their transactions. It's this decentralized nature that makes Bitcoin impervious (so far) to government meddling, free of regulation and monitoring.
Before anyone can even use a Bitcoin, those coins must be mined, by a so-called bitcoin mining process. Any computer can begin mining for bitcoins by using a free mining application. Mining requires the entire network of Bitcoin-participant computers to do a set amount of work before being rewarded with a bitcoin. Basically, that work means a whole lot of number crunching -- and the spoils go to the owner of the computer that completes the set of number crunching at hand. Some people invest many thousands of dollars in very powerful computers just to mine bitcoins, and in essence mining has become a computing arms race, and only those at the leading edge stand to gain anything in the way of profit.
The exact amount of work required is variable. The network adjusts that workload so that the number of bitcoins rises at a steady, predetermined rate. It will continue to do so until the number of bitcoins in circulation reaches its ultimate number, which is 21 million. Currently, the mining process creates 25 bitcoins every 10 minutes. Every four years, the number of coins that can be mined will be halved, until the capped limit of coins is reached in the year 2140. After that point, the number of bitcoins in circulation will be static.
As you already know, you hoard your own bitcoins in a digital wallet. When you send or receive coins, they are verified by a digital signature, called a public-encryption key, which prevents counterfeiting and makes coins recognizable to the network. In a sense, the network shares an open ledger (called the blockchain) of all bitcoins, which provides a trustworthy and redundant way of maintaining the number of bitcoins in circulation.
All of this works thanks to Bitcoin's ingenious open-source (that is, viewable to everyone) code. Open-source software is commonly used by programmers who are opposed to corporate profiteering and control. Any skilled programmer can see how Bitcoin's programming works, and that's OK -- it's not the code that protects transactions. Instead, it's the shared network that verifies the legitimacy of each transfer.