Mining is the heart of the Bitcoin-protocol. The miners are the Accountants of the Ledger which holds virtual/digital-properties. These accountants check and validate if properties are real (not double-spend or counterfeited) and who are the rightful owners.
This is nothing new (you don't need to solve a hard Sudoku puzzle for this) , but what is, is that this Ledger is distributed (not one central ledger nor central accountant) over thousands of places all over the world and this computerized Ledger is updated with tens of transactions every second... so who holds the real Ledger? (the one in Australia is different than one in America). This is solved by a Proof-of-Work protocol: every accountant has to solve a very hard Sudoku-puzzle for which once every about 10 minutes there is an accountant who really solves it first. This account's version of the Ledger will be the one which is the now the only valid Ledger, and all other accountants will honor that (more than 50% admits that the Sudoku is solved) and all the accountants will start to solve another Sudoku for the next 10 minutes.
To incentivize accountants (the Miners) to follow this procedure and keeping the distributed Ledger 100% valid, they get 25 Bitcoins for a solved Sudoku.
Hope this explains it a bit....