bitcoins aren't literally sent to your computer. the transaction occurs on the blockchain, which is on thousands of other local computers.
to view the status of your bitcoin on electrum, export the master public key into your main computer. from there, it is a watching-only wallet and funds cannot be sent out.
This.
It may seem nitpicky but
understanding this concept will save you a lot of confusion in the future. It will also help you understand the whys. Why only you can spend your coins. Why" you don't need to be online to receive coins. Why it is not possible to make fake Bitcoins or fake a transactions.
Your wallet has no coins. That is right it doesn't have a single coin in it. Online offline, QT client, multibit, eWallet I don't care no wallet has any coins.
Copies of the "coins" (unspent outputs from prior txs available for spending), all 12,028,275 as of this post are on every node in the network. Right now, "your coins" are on over a hundred thousands copies of the blockchain, located on strangers computers all over the globe.
So if you wallet doesn't contain coins, what does it contain?
Your wallet has a set of private keys which let you (and only you) spend coins sent to you (or technically to addresses for which you have the matching private key). Your wallet (any wallet) never needs to be online to "receive" coins, because nothing is sent directly to you. When "sending coins", the sender creates a message (transaction) which transfers ownership of "coins" from addresses he control (he has matching private key) to addresses you control (private keys are in your wallet)*. The message is relayed from node to node who all independently verify it and eventually a miner puts it into a block to make it permanent.
When your client (the wallet program) shows a balance of "10 BTC" what is really means is the client scanned the blockchain matching all unspent outputs to addresses which correspond to private keys in the wallet and found one or more unspent output that when totalled are worth "10 BTC". You "have" 10 BTC because you have the ability to send/transfer/spend 10 BTC.
Since coins are on every node in the network the only thing that makes them "yours" if the fact that you (and only you) have the private key to spend them.
In Bitcoin ownership for all practical purposes is control of the private keys to addresses which have unspent outputs.If you lose your wallet = nobody has private keys = the coins will always exist but nobody will be able to spend them.
If an attacker steals your wallet = attack has a duplicate copy of the private keys = attacker can transfer the coins to addresses only he has the private keys for.
* Note that still isn't technically accurate because Bitcoin works on a concept of inputs and outputs, not value or balances at an address. Still this is a technical distinction which isn't really necessary to understand Bitcoin at the user level. Just understand the rabbit hole goes deep. When you think you know everything about Bitcoin ... you don't.
