What's the definition of an "offense against the United States"? Is it just the breaking of a Federal law? What if someone is in jail for a state crime and not a federal crime? Can the President pardon them?

No. Pardons are only for federal crimes. The President cannot pardon someone for state crimes (though often the Governor of that state can).

As far as Biden's decision - man, it's a bad and selfish choice. There's no real reason for the pardon other than that Hunter is his son, and this is something he can give him. There's no evidence that Hunter wasn't given due process, or lacked the resources to fairly contest the charges, or any of the more substantive reasons that the Pardon power has been exercised to correct in the past. Worse, Biden's assertion that Hunter's travails were due primarily to politics - under his DOJ - just gives support to Republican denigration of Trump's prosecutions as mere "lawfare" rather than the application of criminal law.

Biden's legacy had already taken a number of hits, and this is another (completely self-inflicted) one.