Why is mining an important part of a digital coin? I have been playing with the blockchain lately and studying how Bitcoin works. I was wondering though, what is the functional reason for the mining process? I understand that you get a reward for mining, but you could implement that reward also in another way, or do away with it completely.

I first thought it would be to make the blockchain more tamperproof, because the mining will make it extra hard to change a block somewhere up the chain. But, using the signatures and the distributedness this is nigh impossible anyway.

So, why mine?
Why mine?

Money supply

It's a way of limiting the money supply in a decentralised system. You don't want anybody to easily create new money - no-one would trust a currency where anyone can make themselves a millionaire at whim.

Most of the conventional ways of limiting money supply require a central authority who manage the printing of banknotes and their destruction and a regulatory system that prevents banks declaring that they hold more money than they actually do.

Paying for transactions

An electronic payment system consumes processing resources to validate the transactions and to make the appropriate records available to anyone who needs them. There has to be some ways to pay for this.

In a decentralised system, both transaction fees and mining are ways in which this work gets paid for.