What is Bitcoin mining ?

Bitcoin mining is the process of verifying and adding transaction records to the public ledger (the blockchain). The public ledge is called the blockchain because it really consist of a chains of block.

When you make a Bitcoin transaction of 1 BTC to Alice, you would have to broadcast to the world that you are making this transaction. Now that everybody knows that you are sending 1 BTC, what is stopping you from sending that same 1 BTC to Bob, Charlie, and so fourth.

In order to mitigate that, this where mining comes in. Bitcoin mining is deliberately made to be difficult via the 'proof of work', where a complex mathematical problem needs to be solved in order to add a block (which contains that transaction) into the blockchain.

Only transactions that are added to the blockchain are considered final, that everybody can trust that you are not double spending your Bitcoin. As a consumer, all you should care about is that the deeper burried within the chain the block that contains your transaction, the harder it is for an adversary miner/spender to reverse that transaction.

Mining is a highly resource intensive process, hence miners will need to bulk up high computational devices (ie. graphics card, ASIC chip) and also the need to consume a vast amount of electricity to do the 'proof of work'.
To compensate that, miners are rewarded with 25 Bitcoin (as of this writing) which becomes part of the new money creation process (towards reaching the 21 million Bitcoin total supply) for their effort and cost of verifying transactions.

If you would like to see how a "Mining Block" looks like, swing over to Bitcoin Block Explorer - Blockchain.info.
Within each block there is a bulk of transactions
Bitcoin Block #308021 is an example of a Bitcoin block that has been added to the blockchain. It is the 308021th block to be added to the chain by a miner.
If you visit that site, below you will notice a list of transactions that have been included into this block.