What are the Decentralized-Systems Advantages (or Disadvantages) to the Centralized System?

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I'm not gonna pretend I understand this exceedingly well ... I mean; I get 'the end results of the system' (one person buys "money" to send it to someone else, and we 'cryptominers' use the program to add their money to a "blockchain" that gets verified & added-to & verified etc. earning our shares of 'the fee' the buyer paid), but (I think you can see) I don't know exactly HOW we 'add the money to the blockchain.' But then again, I don't know if we're exactly sure how money gets "sent by check." I mean; we all know the basic 'this is my money, and--with these magic codes and my royal signature on the one side and your royal signature on the other--POOF! this amount hath now been added to YOUR account'-routine, but I don't think we really think about 'the bank-workers calling the other bank and verifying the codes and changing the bank-balances & all the other dark secrets of banking' But I think the main advantage of the decentralized system is that it's almost impossible to successfully 'cook the books' (I don't know if that's the right phrase for it). With a centralized system, there's only one 'ledger' that matters (and any ledgers that differ from that one are "wrong" ... more monarchical than democratic, eh?) Decentralized, EVERY ledger matters! How? ... I've still got to learn that (I'm sure SOMEBODY out here knows!) I could look it up, but I don't want to have to earn an Economics Master's to understand it; does anyone here understand it in simpler terms?