I had read about staining a concrete floor, and learned that cleaning it was the important step before staining it.
Accordingly, I swept it, scraped the concrete with a knife, and then vaccuumed up the piles of dirt that were left behind from framing, drywalling and painting.
I mopped once, and the floor looked dirtier than it did before I started. After seeing how dirty the floor was, I figured that I had better do a much better job the second time. I mopped it again a second time. It looked cleaner, but there were still massive mop streaks and arcs of of dirt left behind.
Earlier that day, I had inquired about having the concrete company return to plug in the divits that are left behind by the blossoms from the cherry trees. They were able to come the next day, but said that the concrete would take a week to cure. Unfortunately, the project schedule demanded that I have the concrete stained right away---we can't install cabinets until the concrete floors are stained. And we can't have the final plumbing done until the cabinets are installed and the counter-tops put into place to hold the sink.
So, after mopping a second time, I got down on my knees and carefully scraped the floor with a razor blade, un-caking minute layers of dust and grime. Then, I mopped it again and vacuumed up the water with a shop-vac.
Then, I got on my hands and knees with a sponge and scrubbed every inch of the concrete.
In the morning, cleaned once again, and the proceeded with the staining. Here's a video the staining process.
After two layers of stain throughout the day, on the following day, I sealed the floor, and now will let it dry and cure for 72 hours. I think it's going to turn out very well.