Double check all measurements to ensure a square room. Use a chalk line to mark the inside edge of the planned wall onto the concrete basement floor. Make sure to clearly mark the location of any doorways or openings.
Walls are built from the ground up. Cut pressure treated 2" x 4" boards to length and lay them along the chalk line. Use a hammer drill with 3/16" masonry bit to drill a hole through the wood and into the concrete floor. Use an impact driver to drive a 3" Tapcon screw through the wood and into the floor. Repeat at the other end of the board. Repeat every 16" along the bottom plate. Repeat for remaining bottom plates.
How the top plates are installed depends on which direction the bottom plate is running in relation to the floor joists above. If the joists run perpendicular to the bottom plate then nail the top plates to the joists where they intersect. Where the bottom plate runs parallel to the floor joists, attach 2" x 4" blocking boards between the joists for something to nail into. Make sure the top plate is perfectly aligned with the bottom plate before nailing it to the joists or blocking. Repeat for remaining top plates.
Wall studs are typically installed at 16" intervals when measured at their centers. Start at one end of a wall and mark the location of the first stud. Use a tape measure to mark out the location of the remaining studs along the wall. Using a plumb bob or laser level, transfer those locations to the top plates.
Measure and cut 2" x 4" boards so that they fit snugly between the wall studs. Attach them around the room about 4' from the floor using 3" nails. It is okay to stagger the placement of these blocks to facilitate nailing.