As mentioned, on-chain transactions are done entirely on-chain. The history of the transaction and all its details are available on the blockchain. However because transaction fees and gas costs are dependent on the complexity, this can be somewhat expensive. ​ Off-chain transactions are done through some other means, such as another blockchain or a trusted entity. At some point, the results of the off-chain transactions may eventually be recorded back on-chain as its final outcome. ​ Consider people trading ETH on a centralized token exchange. The ETH started onchain, and was transferred to the wallets held by the centralized exchange. Users of the exchange then buy and sell on the exchange, where the results of each transaction are recorded in the exchange's database. At some point, the current holder may withdraw their ETH back to the blockchain again. ​ From the blockchain's point of view, ETH got sent to a wallet, and likely a different amount got sent back later. All the transactions in-between that happened are not known. And because of that, for all of those trades in between, gas did not need to be paid to the blockchain.