There are two aspects to mining. The first is functional aspect (what is it supposed to achieve) and the second is the technical aspect (what is going on under the hood).
For the sake of clarity I'll deal with both.
In the (very) long term the process of mining is about approving transactions and making them unchangeable once a block of transactions has been included into the blockchain. The Blockchain is a ledger of all transactions that has ever occured. A block is just a group of the latest transactions that when grouped together meet certain technical requirements.
As a reward for doing this (again in the long term) a miner can receive the transaction fees.
In the short term, and as an incentive for early adopters, the miners are given an additional bonus: they receive a reward of 25 bitcoins for "discovering" a block.
OK this is where it gets technical.
The process of block "discovery" is where a mining rig is required to run through a repeating process called "hashing". In laymans terms - hashing is jumbling and mixing up some text in a such a way as to arrive at a repeatable outcome.
In other words if somebody else was to perform the task with the same starting point, they would also arrive at the same end point: the same sequence of jumbled letters and numbers.
So what are the miners looking to discover? They are looking to discover (or more specifically be the first to discover a hash result that starts with a series of zeros.
It is a lottery and in order to do that the miner must use a combination of things: the transactions it is receiving, the previous block are just two, and put that information into it's hashing machine and check each result to see if it matches a hash with the right number of zeros.
Once they have done that - they publish the result to the network, and every other node checks the results for correctness. The result is easy to reproduce once someone knows what is required to go into it.
There is a third thing that I didn't mention - equally important but easy to grasp: the miner must check each transaction against its own copy of the ledger to validate that each transaction is genuine.
Obviously if you want to win this lottery computing power is required, therefore more and more specialised equipment is needed. An ASIC is a chip that only does one task: hashing. Therefore it can be designed in such a way as to perform the task as fast as physically possible, and for this reason it has a much better chance of winning the reward of the 25BTCs and the transaction fees for all the transactions in the block.