On-Chain transactions refer to cryptocurrency transactions that occur on the blockchain and remain dependent on the state of the blockchain for their validity.
All such on-chain transactions occur and are considered to be valid only when the blockchain is modified to reflect these transactions on the public ledger records.
A transaction is a transfer of value in a particular cryptocoin token, the details of which are recorded on suitable blocks of the blockchain, and the same are broadcast to the whole cryptocurrency network after suitable verification.
Depending upon the network protocol, once a transaction garners sufficient confirmations from network participants based on the network's consensus mechanism, it becomes almost irreversible. (It can only be reversed if the majority of the blockchain's hashing power comes to a consensus to reverse the transaction.)