Say someone sends me 0.5 of a BTC to my desktop wallet, is it a digital file or lines of code that they send me which has monetary value? I'm very confused on what you actually RECEIVE with cryptocurrency.
Think of Bitcoin like a ledger. When you (your Bitcoin wallet) receives Bitcoin, instead of receiving an item or file, everyone is updated with an updated ledger including the balance of your Bitcoin address.
Here is an extraordinary explanation on Bitcoin.
"you receive a bitcoin" means that someone makes a transaction with your BTC-adress as aim. Nothing of this happens on your computer or in your wallet. It's everything in the decentralized blockchain.
Your wallet is only an easy tool to save you private key, to ask the network to make a transaction, to see your total amound... And with the private key, you have access to the adress where someone sent the bitcoin to.
Hopefully this analogy helps clear it up:
Your bitcoin address is like your bank account number and your private key is like your bank account pin. Then the blockchain is a ledger just like if you were to log into chase online banking. But bitcoin is decentralized and chase is centralized.
It's much the same as when you have a dollar. When you have a dollar, you have the ability to give someone else a dollar, no more, no less. When someone sends 0.5 BTC to your account, what they've done is give your account the ability to transfer 0.5 BTC to someone else's account.
When your bank agrees to give you a loan, and creates an account for you with a $10,000 balance, which you can use to pay bills from your computer, what do you actually get? (Would have been a comment if I'd had sufficient rep.)