An example of how bitcoin works on a bit-level: Ok, I'll give it a shot.
Here's what the current best-block (according to my bitcoin client) looks like, dumped in a geek-readable format:
BLOCK 68fa61ac1f55a5787dfa0c75bc83e67376ae8356e6887a2ab74cdb0900000000
Next block: 0000000000000000000000000000000000000000000000000000000000000000
Time: Mon Jul 5 15:51:22 2010
Previous block: c18adb50289393b5a995b3506f039ac75e8de79f511515448811510200000000
3 transactions:
1 tx in, 1 out
['TxIn: COIN GENERATED coinbase:0442310d1c029c00']
['TxOut: value: 50.00 pubkey: 17sdrb1X7qpjPMJortqaNwWtBbtouSoJn2 Script: 65:046d...bb9c CHECKSIG']
1 tx in, 1 out
['TxIn: prev(580a...e82e:0) pubkey: (None) sig: 71:3044...db01']
['TxOut: value: 50.00 pubkey: 1FeFgJRvCYUTCBj1u696eL23xpAdNB4B8p Script: DUP HASH160 20:a09d...6d81 EQUALVERIFY CHECKSIG']
3 tx in, 1 out
['TxIn: prev(c0a0...6bc3:0) pubkey: (None) sig: 73:3046...0f01', 'TxIn: prev(f909...2493:0) pubkey: (None) sig: 73:3046...1601', 'TxIn: prev(bc0a...fe64:0) pubkey: (None) sig: 72:3045...6201']
['TxOut: value: 150.00 pubkey: 1BHxjkqPmtNdmJxLZgneijvGszRxM9hPkz Script: 65:04ee...1d02 CHECKSIG']
So: that big long string of hex at the top is the block header's hash value. Note that it ends with 8 zeroes; that's the proof-of-work (my utility for dumping blocks doesn't bother dumping the Nonce values).
What's hashed in the block header? The Nonce. The block's generation time. The previous block's hash. And a hash of all the transactions in the block. (and probably some stuff I'm forgetting).
This block has three transactions in it. The first is the 50.00 (which is really 5,000,000,000 of the smallest possible units) reward for finding/creating the block. It can only be spent by whoever has the private key that matches the public key in the TxOut (17sdrb1X7qpjPMJortqaNwWtBbtouSoJn2 -- you can think of public keys and bitcoin addresses as equivalent), which will be whoever generated the block.
The second is a payment of 50.0 from.... somebody... to... somebody. How does Bitcoin know that transaction is valid? Well, it:
+ Looks up the previous transaction. That's the TxIn: prev(580a...e82e:0) stuff-- fetch TxOut zero (which will be a coin generated txn) from previous transaction 580a....
+ EVALUATE(TxIn.pubkey + previous transaction TxOut.pubkey) and make sure it evaluates to true. This is where the cryptography happens; the receiver uses the private key known only to them and provides a correct digital signature.
The third is a payment of 150.0 (three 50.0-value in, one 150.0-value out).
Clear as mud?