Bitcoin mining is the processing of transactions on the Bitcoin network and securing them into the blockchain. Each set of transactions that are processed is a block. The block is secured by the miners. Miners do this by creating a hash that is created from the transactions in the block. This cryptographic hash is then added to the block. The next block of transactions will look to the previous block’s hash to verify it is legitimate. Then your miner will attempt to create a new block that contains current transactions and new hash before anyone else’s miner can do so.

Though transaction fees are optional, miners can choose which transactions to process and prioritize those that pay higher fees.[67] Miners may choose transactions based on the fee paid relative to their storage size, not the absolute amount of money paid as a fee. These fees are generally measured in satoshis per byte (sat/b). The size of transactions is dependent on the number of inputs used to create the transaction, and the number of outputs.[3]:ch. 8

How hard are the puzzles involved in mining? Well, that depends on how much effort is being put into mining across the network. The difficulty of the mining can be adjusted, and is adjusted by the protocol every 2016 blocks, or roughly every 2 weeks. The difficulty adjusts itself with the aim of keeping the rate of block discovery constant. Thus if more computational power is employed in mining, then the difficulty will adjust upwards to make mining harder.  And if computational power is taken off of the network, the opposite happens. The difficulty adjusts downward to make mining easier.
In January 2009, the bitcoin network was created when Nakamoto mined the first block of the chain, known as the genesis block.[18][19] Embedded in the coinbase of this block was the following text: "The Times 03/Jan/2009 Chancellor on brink of second bailout for banks."[10] This note has been interpreted as both a timestamp and a comment on the instability caused by fractional-reserve banking.[20]:18