Sorry for coming on here and whingeing about the time it takes to synch.
No problem. Everybody is new to bitcoin at some point.
But say I want to keep an account ready with some money for small expenses online. Like paying this usenet site that I mentioned, paying a membership for a club and a bit of this and that.....
To achieve that, I want to transfer money onto an "account". Is the account the Wallet (parent in the relationship) or the Address (child in the relationship).
Say my wallet is called "Freja's wallet".
blockchain says my address is: 1ApeaRCTBuPStVrA1zK2d3xTM5VmDCTaVj (not real address)
Where is the money/value stored? I thought it was in the address?
Hmmm, a lot of technical understanding will be needed to grasp the details you are asking about. Lets start with saying that the value is all represented in the blockchain. Your wallet just contains the private keys (perhaps think of them as passwords) that allow you request blockchain updates that show that value as associated with someone else's address.
So, If I send 1 BTC to address 1ApeaRCTBuPStVrA1zK2d3xTM5VmDCTaVj. I send a transaction out into the bitcoin network indicating that 1 BTC worth of the value that I control is no longer under my control, but is now to be considered under control of whoever has the private key to that address instead. My bitcoin client signs this transaction with the private key from my "wallet" (in the case of the reference client this is a file named wallet.dat) and everybody in the entire network can verify that signature against my previous ownership of that value. At that point nobody (including me) can spend that particular 1 BTC of value (lets ignore double-spend attacks for now) unless they have the private key that is associated with the new address that my transaction indicated. If nobody has the private key, then that 1 BTC will never again be spent (so losing a wallet.dat in a disk crash means permanently losing access to your bitcoins unless you've backed it up).
Note that you don't have to be running your client at the time I place the transaction for it to succeed. If you ran a program that knew the proper format of a private key and how to mathematically generate a bitcoin address from that value, you could tell me the bitcoin address and print out the private key on a piece of paper. I could still send my transaction into the bitcoin network indicating that a signature from a private key associated with the address is needed to re-spend the value, even though the address you gave me has never been seen on the bitcoin network prior to my transaction. Then at a later date if you could find a way to import your printed private key into a client you could spend the value. (This is what people are talking about when they talk about "paper wallets")
Since your client generates your addresses, it also generates the necessary private keys and stores them. This means that (unless you share the private keys hidden in your wallet - or they are stolen) you are the only person who can create a new transaction that uses any particular value that has been transferred to an address your client creates.
Eventually a miner adds these broadcast transactions to the blockchain, and there is a permanent record of the transfers having happened.
So, you don't really have an "account". What you have is a set of "addresses" that you can give to others so they can send transactions containing value into the network that you can forward on to others with your own transactions using signatures associated with the addresses. The client takes care of keeping track of which signatures are needed when creating a new transaction and keeps track of all the transactions in the blockchain that indicate value transferred to you. When you see an indication in your client of the amount of "bitcoins" you have, what the client is doing is adding up all the unspent value in all the transactions on the network (and in the blockchain) and showing you the total.
You can create as many addresses as you like. There are even programs out there that will continuously create addresses as fast as possible and check to see if the created address matches a pattern (such as starting with 1Freja...) throwing away all the addresses that don't match. When it finally finds an address that matches the pattern, it will give you the private key so you can import it into a client and have your own "vanity address".
So the idea behind a unique addresses for a transaction is generally suggested when giving out an address for someone (or some program, or some currency exchange) to send you bitcoin. This way the person (or people) who now know one of your addresses (since you just gave it to them) don't know what other addresses you have access to and therefore don't know how much bitcoin value you control or where you are sending bitcoin value that you've received from others. It offers a small amount of token anonymity. If you are truly concerned about privacy and anonymity you'll need to do a lot more than that.
. . . say that I become suspected of terrorism and the police turns up and seizes my computer.
Assuming that you have all your bitcoin addresses on your computer, at this point it is too late. They have (from your computer) a list of ALL your addresses and therefore can trace every bitcoin transaction you have ever made (assuming you haven't deleted any addresses/private keys from the client).
Here is an example where anonymity could come in handy...
Say you run a blog or YouTube channel or some other situation where you list a bitcoin address for anyone to send you some bitcoins if they like the content you offer. Clearly this address is publicly known. Now say you have converted £10,000 into bitcoin at an exchange and transferred it to the same address. You decide you want to make some sort of purchase, and are engaged in negotiations with a seller. The seller seeing the address from your YouTube channel (or whatever) knows exactly how much bitcoin you have available to spend. This gives him an edge in the negotiations.
Using bitcoin without knowing how to carefully maintain complete anonymity will still be vulnerable to forensic accounting processes and you should assume that the police can track down your identity if they have reason to target you. If that level of anonymity is important to you (even if you are doing nothing illegal and just value your privacy), then you'll need to learn quite a bit more about obtaining, securing, and spending bitcoin anonymously.
They figure out that my address was the one I gave above. I had not been following the advice you gave, but used the same address for everything. They could then (if they were clued up) read my financial history as easily as if they had my Visa statement.....?!!
Well, they could determine exactly what addresses you sent bitcoin to (or received from) and how much you sent to (or received from) each address, but they'd still have to do the legwork to try and determine who has the private keys associated with any of those addresses. Some will be public knowledge or easy to find, other addresses may only have been used a single time by the person you engaged in the transaction with and therefore there is no further trace as to who specifically you gave the money to or received it from.
But the privacy and safety then comes down to human error and/or access to a physical machine or somewhere where the user has saved the data.
It comes down to the ability of someone else to determine who has the private key associated with a particular address. Every time you give out an address for someone to send you bitcoins, that person knows that you have the private key associated with that address and knows that any other money sent to or from that address was sent to or from you.
