I'm new to bitcoins and I have an online wallet at blockchain.info, but I want to understand wallets in general. How do I manage an offline wallet? I've read some terms like paper wallet, brain wallet, what are they?
Concrete question: suppose I create a wallet that's kept on my disk, how does money flow into it and out of it? Do I need to be online for that to happen, does the wallet file change as money goes in and out, or is that all kept virtually in the blockchain? If the latter, how do I find out how many bitcoins I have in the wallet?
Question 1) To manage an offline wallet, you create it with your internet turned off (preferably using a linux live-cd) and then print it or write down the private key and address (information which references a spot on the blockchain or ledger that everybody keeps which is where your bitcoin is actually stored). To generate one go to
www.bitaddress.org, then drop offline (maybe unplug ethernet), and generate your information, print it, or write it down, then turn off your computer before going back online. This is best done from a linux live cd like ubuntu.
Once you send money out of your offline wallet it becomes compromised and is no longer an offline wallet. Also, if you load your private into a wallet managing app (like the bitcoin client, or blockchain.info) make sure you backup the wallet.dat, or send all of your bitcoin to a new wallet in one transaction. This is because of a quirk of the bitcoin clients called "change." Also, do not do this with Litecoin yet.
question 2) paper wallet is just a private key written down on a piece of paper. brainwallet is where you come up with a secure password that you can remember, and use an algorithm (bitaddress.org, again) to link it mathematically to one specific key. That way, all you need is the address to put money in it, then you can have money stored in your brain, only accessible if you provide the info.
question 3) it doesnt. its stored as information written on a ledger, kept by thousands of people, and verified cryptographically by miners. The wallet app is just a means of accessing and changing the blockchain.
therefore question 4) Yes, you do need to be online to send a transaction. Unless you have a spare old computer and set up a program called "armory" your offline wallet will be compromised when you go to send money out of it. You do not need to be online to generate a private key to a spot on the blockchain, and send money to it, however-- hence the concept of offline wallets. Yes, the wallet.dat file changes. I don't think you have to keep the changes though, just a backup. Not sure about that last sentence though, so I update it everytime just to be safe.
Question 6) already answered
Question 7) Use a blockexplorer. Blockexplorer.com comes to mind, but there are others i think. Once you send money out of a wallet blockexplorer might get messed up because of change.
Hope this has been helpful.