Bitcoins, unlike paper currencies, don't physically exist anywhere. One can talk about having bitcoins but you can never point to a specific object, a physical entity to be exact. It is not even a digital file. Instead, there are only transaction records. If you already have bitcoins after going through the set up and reading all the resources from Bitcoinmining.com and you want to make a transaction, this is how these transactions work.
A bitcoin transaction is sent to and from a bitcoin wallet. They are also digitally signed. Everyone knows the transaction and its history can be tracked starting from where the bitcoins came from. There are records between different bitcoin addresses with the balance amounts. Every transaction is stored in a blockchain or a public ledger.
There are 3 pieces of information to a transaction: an input, an amount and an output. If person C receives some bitcoins from person B, the input is the record of the bitcoin address that sent the bitcoins to person B in the first place. An amount is the amount of bitcoins sent to person C by person B. An output is person C's bitcoin address.
A bitcoin address and private key are needed in order to send bitcoins. A bitcoin address is a sequence of random letters and numbers. A private key is also a sequence of letters and numbers but is kept hidden. A bitcoin address is public while a private key is private.
So, when person B sends bitcoins to person C, person B signs in with his private key and specifies the input, amount and output. After this, he sends the bitcoin wallet to the network. Miners then verify the information, put it into a block and solve it eventually.
A transaction usually clears up after 10 minutes since miners need to verify the transaction although some transactions can be confirmed right away. There are merchants that do not let you wait. This usually happens for transactions that are of low value.
There are sometimes transaction fees but it doesn't happen always. There are also no receipts.
If you need further reading, Bitcoinmining.com just might help.