When in dealing with blockchain environments, Computation is extremely costly and slow, however, using a blockchain ledger for proofing data is extremely efficient and reliable; therefore, one of two things can take place: 1.) either the data exists within the blockchain itself (On-chain) or 2.) a hash reference of that data exists on the blockchain and the activity is conducted elsewhere (Off-chain).
The Term, On-Chain, is used to refer to any transaction/event that is processed and registered within a blockchain.
On-Chain & Off-chain principles can be more easily understood when they are thought of as online/offline. You can use your computers Microsoft Word without the internet so that activity is off-line (off-chain). However, if you use google docs you must be connected to the internet (online/on-chain). In the first example only you know about the activity that happened; meanwhile, in the latter example, Google registered all of your activity on their servers.