Start out by filling your bucket or kitchen sink with hot water and some kind of cleaning solution. If you have a favorite, use that. A little dish soap and some vinegar is great just as well too!
Next, pick up your mop! Despite all the new mopping gizmos out there, any kind of cotton mop is still the best. It takes in water and dirt, but it also lets it go. The great thing about these types of mops is that you can remove the mop head or pad off and wash it either in a washing machine or by hand and then dry it out real well. With some of the foamy mops, you never really know if they're clean and you end up having to replace the heads more often because they get gross, smelly, and they spread the dirt around.
You need to be able to twist and wring out your mop head or pad. Move your mop back and forth over a small area. Rinse and wring out.
Go back with your freshly wrung-out mop and pick that dirty water back up off the floor!
Only swish your mop back and forth about three times over that spot! Then wring it out and if there's still more water to pick up, do it again! If you go over it more than three times, you're just putting the dirty water right back down!
So then you just go all around your floor, putting clean water down and then picking it back up a few seconds later. In no time, your floor will be the cleanest it's ever been!