The 'equation' is finding a hash that is less than a certain value, which is called the difficulty, and is dynamically adjusted so the approximate amount of time it takes to calculate the next one is 10 minutes.
A hash, in this case, is a number whose bits are tied to as many bits of the input as possible and thus are a more or less unique identifier of the content. A super-simplistic and not at all good hash is a simple crc where you add each byte of the input to the next. If you had the bytes:
the hash of that input would be 1 + 2 + 3 + 4 = 10.
The hash that bitcoin uses is called
sha256 and results in a 256 bit number that has every bit of the input affecting every bit of the hash. It is much more complex and much slower than the algorithm above.
What happens in bitcoin is that when someone wants to send some of their bitcoins to someone else, they sign a transaction with their private key (more on this at
http://en.wikipedia.org/wiki/Public_key_cryptography), saying which public key it's from and to and how much it is and then they broadcast it over a network that works in some ways like bitttorrent. Every other bitcoin client gets a copy of this.
Miners take a set of these transactions, put them into a data structure, which also includes the hash of the last block added to the block chain and a field called the nonce, and then hash the structure (called a block). If the hash results in a number lower than the current network difficulty, the block is broadcast and other clients add it to their block chain and it, and all the transactions in it, basically becomes permanent at that point. If not, the nonce is incremented and the block is hashed again. Repeat until either a new block has come online or a block has been found. The miner also has the privilege to be able to add an arbitrary transaction (or set of transactions) into the block that outputs an amount of bitcoin to their public key (or whatever public key(s) they choose). This amount was, for the first few years of bitcoin, 50btc. It halves after a certain number of additional blocks and is currently 25btc, and will next drop to 12btc in a couple years or so. They also get to output to themselves fees that people creating transactions have attached to their transactions to encourage them to be picked up by miners. When the reward for block discovery reaches zero in a couple of decades it's expected that these fees will be the main reason to continue mining.
Every 2016 blocks the difficulty is adjusted. It either moves up or down depending on how long it took to generate those 2016 blocks. The difficulty is calculated such that it is a number sufficiently low to cause the next 2016 blocks to be generated in about 2 weeks. This is calculable by any client on the network from the timestamps on the blocks in the block chain that everyone's agreed on.
The reason all of this actually works and guarantees the validity of the transactions is that an immense amount of computing power would be required to put arbitrary blocks into the block chain. The effort is distributed over many many entities that all must agree on everything, and the only way to completely arbitrarily usurp the power is to have more hashing power than the rest of the network combined, because otherwise your bad blocks will be ignored and displaced by good blocks from good actors.
Whatever you think about bitcoin as a currency or store of value, it is an immense technical achievement and an extremely well designed system. Understanding it should be required for anyone who programs networked systems at this point because it's a huge leap forward on that front.
