I keep seeing the words offline forging and forged mentioned in various Tezos blogs and tutorials.
What is the purpose of forging?
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From a high level perspective the purpose of forging is to convert a set of inputs along with the kind of operation you wish to perform (eg: send money, to whom, from whom, how much etc) into a single sequence of bytes which will uniquely represent said operation.
Then the next step for this operation to enter the chain is to sign those bytes with your public key in order to prove publicly that you are the author of such bytes. The output of this will be another sequence of bytes.
Eventually those bytes will be actually pushed to the chain by an injection step.
All of those are performed by various clients/wallets behind the scene when you perform operations.
More details are available here