When someone sends bitcoins to your bitcoin address, the transaction is broadcast to the bitcoin network and added to the blockchain.  The transaction is confirmed by the nodes of the bitcoin network and subsequently included in the blockchain.  When the Electrum application is launched and connects to the bitcoin network, the application checks the blockchain to see if there are any incoming transactions for your bitcoin address.

 If the device is turned off, the Electrum application cannot be running to receive incoming transactions.  However, as soon as you turn on the device and launch the Electrum application, it will be synchronized with the blockchain and will check if there are any incoming transactions for your bitcoin address.