Install a wooden subfloor to help insulate flooring from cold concrete.
You can install laminate, engineered wood, resilient flooring, and many other finished flooring materials directly over a dry concrete slab. However a wooden subfloor helps insulate the flooring from the cold concrete. It also adds a little cushion to the hard surface of the concrete. A wooden subfloor will not cure moisture problems. Moisture that comes up through the slab will ultimately find its way into your finished floor. Cure the moisture problem before you install the floor.
Headroom, especially in a basement, may be at a premium. To make low flooring framing that will conserve headroom, lay pressure-treated 2x4s (sleepers) flat and cover them with 3/4-inch tongue-and-groove OSB underlayment or exterior plywood.
Prepare the slab by vacuuming it, then seal with an asphalt primer and a layer of asphalt mastic. Lay a moisture barrier (6 mil polyethylene is often used) and mark the plastic sheet with a felt marker so you can position the sleepers on 16-inch centers.
About 8 hours for a 10x10 floor
Circular saw, level, powder-actuated gun, cordless drill
Measuring and marking, cutting lumber, installing mastic and waterproofing membrane using powder-actuated fasteners
Clean and repair floor
Polyethylene membrane, tape, powder-actuated fasteners, shims, screws, 3/4-inch tongue-and-groove OSB underlayment or exterior plywood, pressure-treated 2x4s
Prepare the floor by vacuuming and installing a waterproofing membrane on asphalt mastic. Overlap the seams of the plastic sheet at least 6 inches and tape the seams with the tape recommended by the manufacturer. Starting with a 1/4-inch gap along the walls, mark the sheet with a felt marker at 16-inch intervals. Snap chalklines to help you position the sleepers. Set out the sleepers along the perimeter of the room and in their approximate locations across the rest of the floor, cutting them to length as necessary.
Set the first sheet of underlayment (tongue to the wall) with the long side perpendicular to the sleepers and 1/2 inch from the wall. Fasten the sheet with 2?1/2-inch coated screws every 6 inches along the walls and every 8 inches across the sheet. Don't drive screws at the grooved edge yet.
Install the remaining sheets in the first row along the wall with a 1/8-inch gap between the ends. In the second row start with a half-sheet (then a full sheet for the third row) so the joints will be offset. Fit the tongue of one sheet into the groove of the other and screw it to the sleepers.