I have accidently tunneled stairs into a small amount of underground water, and now my main hall has 1/7 water on the floor, and the stairs below are completely flooded, how can I remove it?
1/7 water will go away on its own after a little while. However, clearing out the actually flooded area is going to be harder.
First, make sure there isn't more water flowing in, especially from the edge of the map (called sourced water). If there is, that's the first problem to fix. You may be able to use pumps to drain it off fast enough to build a dike, but it's pretty aggravating. If your stairway is flooded with sourced water, it might be best to write it off as a bad job, wall the whole thing off, and start digging anew somewhere else. Be careful not to wall any dwarves in. (For the love of God, Montresor!) If you really want your stairway right here, fortunately you're already underground, so you should be able to block the water with a controlled cave-in, much like you would to pierce an aquifer.
Assuming the flood is stable and it's not sourced water, you have a few choices. The easiest (but most labor-intensive) is to designate a water source zone next to (or around) the flooded area, and a pond zone next to a convenient large pit somewhere else on your map. If you do this right, your dwarves will haul the water from the flooded area to the pond, one bucketful at a time (make sure you have some buckets). This will work but it's very very slow.
The next easiest way is to set up a pump next to the flooded area, and pump the water somewhere more convenient. Assuming you have access to the north or south of the area in your picture, you can put a screw pump adjacent to the flooded area but on the level above the water (so the pump should go on the level depicted, probably), with the light square pointing toward the flooded area and the dark square pointing toward some kind of drainage system (a tunnel leading to somewhere you don't care about flooding). You can have your dwarves work this pump manually or you can hook it up to power if you already have power. This should clear out a moderate amount of water pretty quickly.
If you're feeling really intrepid, you can dig around and just breach the bottom of your flooded area, allowing the water to go somewhere you don't care about. This is pretty low effort but pretty high danger. There are some safe-ish ways to do it, described on the wiki.
Finally, if you live in an biome with winter, you might be able to just open the flooded area up to the sky and wait for it to freeze, then dig it out. Make sure you put up proper stone walls before everything melts again.