If you and a recipient both have a wallet on a flash drive, is it then possible to transfer bitcoins from one wallet to the other using a computer without going through the web?
Example in life: "Here you go nephew, I'll give you 5 bitcoins for your birthday." Then we go to the computer with our wallets and transfer the funds, without waiting for people to decrypt bits and pieces of it across the web, just relying on the local computer.
Yes, the simplest example is that you could create a paper wallet, send coins to that wallet, and give the paper wallet to your nephew.
In this sense, the paper wallet act similar to a bearer bond. Whomever holds it, has access to the coins. The caveat is that your nephew has to trust that you don't spend the coins (since you have seen the private key).
No, Bitcoins cannot be transferred to another address without connection to the network.
Bitcoins are not actually stored on a particular device, Bitcoins only exist as claims noted in the network's ledger, the blockchain. What is actually stored on the device's wallet are the means to exert control over Bitcoin claims, the private keys. (For the user's convenience, the wallet also presents information on what the total of that claims pertains to.)
Therefore, to send bitcoins to another address, the state of the public ledger needs to be updated, which can only happen by a transaction being broadcasted to the network and accepted into a block, recording the change in claims of the sender and recipient.
However, bitcoin claims can be transferred by passing along the control over the claims, i.e. by giving the recipient access to a private key. This can be achieved for example by giving the sender an electronic copy of a private key or a paper wallet.
No, this is not how bitcoin works. Bitcoins don't really "exist" per se. Bitcoins are just an abstraction from the public ledger, or blockchain. So basically when you mine, the blockchain has a built in mechanism for rewarding miners. This is where the initial money comes from. When those miners give someone else bitcoins, those people don't "own" the bitcoin in their wallet. They only have control of the bitcoin because the blockchain says that they do. All that is stored in your wallet (mostly) is a public and private key.
There is one way though. If you want to give someone a few bitcoins, pre create a new address and transfer those many bitcoins to it, and when you meet him when there is no internet access, give him the private key of the new bitcoin address. This way, you can give bitcoins even if there is no internet access.
While this is possible it is a lot like a traditional check. For example, if we are both in a starbucks with no internet access (as is the case more and more), and I would like to reimburse you for the coffee you just purchased for me, having a satoshi client (original client), I can create a raw transaction from my wallet.dat file and the blockchain I have saved locally containing unspent coins.
This is similar to a personal check. You can hold on to this signed transaction until you find a location with internet and then "deposit" it to the network. As with personal checks, at least in the US, if I am dishonest and either sign a bogus transaction, or spend the coins before you can deposit your signed transaction, the request will "bounce" and you will not get any coins. At least there's no overdraft ;)
tl;dr offline payments are possible however you should do so at your utmost discretion and only with people you know and trust fully.
More info on transactions and how they work here: https://en.bitcoin.it/wiki/Transactions
