Mining means no one can create a fake block chain (version of events). To create a fake chain accepted by the network someone would have to solve all the hashing problem linking their fake blocks faster than the bitcoin network. The proof of work linking the blocks means they cant be faked.
The advantage nodes in the P2P network (people running the client software not necessarily mining) don't have to trust any other node or central serve, they just trust the longest block chain in he knowledge that's what everyone else will be doing.
As transactions are only accepted if signed by a private key, so the block chain is just an agreement of the order and time that transactions happen. Also if someone tries to double spend the same money by signing two transactions the block chain is the decision on which one is accepted and which ignored. No one can fake that decision without providing the solution to the extremely hard problem.
The advantage nodes in the P2P network (people running the client software not necessarily mining) don't have to trust any other node or central serve, they just trust the longest block chain in he knowledge that's what everyone else will be doing.
As transactions are only accepted if signed by a private key, so the block chain is just an agreement of the order and time that transactions happen. Also if someone tries to double spend the same money by signing two transactions the block chain is the decision on which one is accepted and which ignored. No one can fake that decision without providing the solution to the extremely hard problem.