Using Bitcoin as an example, your currencies are going to be held in a digital wallet. This wallet is recognized by its “public key” which is a unique blockchain address that anyone can view. A “private key” allows the user to request a transaction on the blockchain.
When a transaction is initiated, it becomes pending (as any transaction would). This is almost instantaneous. The transaction is packaged with other pending transactions thereby creating what we call a “block” of transactions. Blocks are files where bitcoin network data is permanently recorded. A block records recent bitcoin transactions. The verification of these transaction is where it gets exciting. There are participating computers (Miners) that run a current copy of the blockchain at all times. They are there to find new blocks, as well as verify the accuracy of all transactions in each block. After a block of pending transactions is created, the miners race to confirm the validity of it.
Once one miner with enough computing power uncovers the newest block, consensus is agreed that the block is valid, it receives a very secure “time stamp” logging the details of that transaction. Once bundled and verified, this block is embedded into the blockchain, and the first miner to discover it is rewarded in ‘X’ amount of new bitcoins for uncovering the block. Each block builds on the previous block like a “chain” , except that every time a new block is mined (added to the end of the chain), the security of all blocks which were mined before it increases. Every block created makes it dramatically harder to rewrite history because anyone can verify that the history is correct when looking back in time to the very first block ever created.
All of this happens within minutes, as opposed to days in a traditional bank transfer. There is no trust needed because the blockchain is not centralized - meaning there are no bottlenecks and no single points of failure.
For a more technical description of the process, here’s an illustration: