Why does Bitcoin mining exist?
A Bitcoin is just a currency. In its most simple form, it is not different from the dollar, Euro, Pound, or whatever, that you happen to have in your pocket. People exchange them for goods and services. However, instead of a physical coin or bill, it is all digital. Like a paypal balance or Steam Credit or even really a debit account.
Making Bitcoin cheat-proof
People are very good at cheating computers. Bitcoin, however, is used as real money. If people could cheat the system to generate Bitcoins falsely, people would lose faith in it, and not use it as money. To prevent people from cheating the system, every transaction you make is encrypted with a rather complicated mathematical formula. The exact formula isn’t important.
Verification of transactions
“Mining” is a bit misnamed when you are going for understanding this principle. Computers that are “mining” check the Transactions that other people have made to see that they are real, valid, and actually happened. When a person makes a transaction with Bitcoins, they are subject to a small transaction fee. The “Miners” that verify that the transaction was real get this fee. In addition, Miners that complete verifying a certain set of transactions (Called a block) are rewarded with new Bitcoins that nobody has ever owned before.
When the new Bitcoins eventually run out, the “miners” will still get the transaction fee for verifying transactions. However, they will not be rewarded with new Bitcoins, as there are no more new Bitcoins to give.