This is a tricky question. Cryptocurrency is virtual, so it isn’t physically stored anywhere. However, what is stored is the transaction data of the currency. The public blockchain is a list of records that can receive new transaction records from anyone with access to the internet. Any trades or coins mined are stored on this blockchain. When you send cryptocurrency to someone, that transaction request is evaluated against the blockchain. Every trade and mining is totaled to determine how many coins you have (virtually) and whether it is enough to process your request. If it checks out, another block is added to the blockchain with your new transaction. If you had fewer coins than you wanted to send, your request would be rejected. On average, the blockchain takes about 10 minutes to send its first confirmation and an hour to confirm the full request.
So how do wallets fit into all of this? A wallet, whether owned through an exchange (an entity that allows trades) or privately stored on your computer, contains the public key and the private key you need to access your coins. When someone sends you cryptocurrency, they send it using your public key. You can then access that cryptocurrency using your private key. The two elements define your claim to those coins. To understand public and private keys, think of a mailbox that has a door and a slot with key A to open the door and key B to open the slot. Anyone that has key B can deposit items into the mailbox, but only the person with key A can access what is in the mailbox. Key A behaves as the private key and key B would be the public key. If you lose your private key, you will have lost access to what is in your mailbox - in this case, your cryptocurrency. If your private key gets into the wrong hands, they will now have access to what is in your mailbox and can take whatever they would like. Due to this, if you lose your private key, recovery isn’t an option. The private keys generally aren’t stored because of the possible risk of it getting hacked.