Great replies, I'm starting to understand. That analogy above helped a lot. One of the final things I'm trying to understand is the coinbase transaction (What I referred to above as something like the new bitcoin generated from mining). The coinbase transaction, being as I interpret it is the original moment of bitcoin "accumulation" by the mining address. This doesn't have the input from the preceding electronic signature because there was no previous owner, just the "output" to the mining address that "earned" the new Bitcoin. When the first owner of the new Bitcoin transfers it to another address, how does the input value get determined(When Bitcoin goes from non-existent to new address, how is sign. defined without "nothing" and new address matching public keys)? Does maybe the origination point have a special first public key? I looked at The White Paper but Satoshi delved through everything really quickly and went over my head a bit.
The block chain keeps track of "transactions" rather than "coins". Transactions have "outputs", and outputs have definite amounts.
When a miner creates a coinbase transaction, his coinbase has one or more outputs and amounts. Anything is acceptable as long as the total amounts don't exceed the reward amount (plus any fees earned). So, his coinbase could give the entire 25BTC reward to a single address, or could split it among ten addresses in ten different amounts as long as the total doesn't exceed 25BTC.
A transaction output can only be spent in its entirety. When an output is spent, if it's "too much", the new spend must contain another output that returns the change back to the sender. If it's "not enough", the transaction can spend multiple outputs, effectively joining them together.
Actually, one more thing. Since a Bitcoin can be divided up into 100 million parts(satoshis) if I have, say, 1 bitcoin with a specific electronic signature, what happens when I spend 1/2 of the Bitcoin? Does the the electronic signature of my retained 1/2 Bitcoin change to verify it's of lesser value? Or does each of the satoshis have a represented block? And if I have 1/2 Bitcoin in my wallet and get another 1/2 will this combine to be one signature? So far from learning about it I have the impression that this system is complete genius.
The amount itself doesn't matter, a transaction output for 0.00000001 bitcoin looks much identical for one for 10000 bitcoins, the only difference being the amount field.
If you have 0.5 BTC in your wallet (assuming that's one transaction output) and you receive another 0.5 BTC, you will have (at least) two transaction outputs available for spending.
If you spend any amount up to 0.5 BTC, your wallet will consume one of those, sending you back any change.
If you spend any amount above 0.5 BTC and up to 1 BTC, your wallet will consume both of those, sending you back any change.