Quarry mining is a very simple way of mining, you just pick an area. So, 25×25 or 30×15, or whatever you want. Then you mine that area all the way to the bedrock, directly down taking all the blocks away as you go.
This is a very easy way of mining, so I don’t really have much to talk about here. Just make sure you build a staircase all the way back to the top, or use ladders, whatever you prefer. I use cobblestone as a staircase. This is a very quick way of gathering a lot of simple building blocks, (mostly Cobblestone).