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It refers to transactions occuring outside the blockchain and executed instantly. There are number of methods to do off-chain transactions. For example, two interested parties do a transfer agreement. Next, a third party faciliatete the transaction by become the guarantor in it.
It is number of transactions done per second. For example, there are 10 transactions of Bitcoin done in 1-minute. The TPS would be 10 transactions/60 seconds = ~0.17 TPS.
