A half wall is easy, use 2*4's, cut a top and bottom plate the desired length of your wall. measure your desired height and cut studs that length minus the top and bottom plates. i would recommend screws instead of nails since you will only be tying into the wall and floor for support.
You must have a base support, like a existing wall you are connecting the half wall to. Lay and anchor a 2x4 that has been cut to length along the floor. Mark how high the wall will go. Cut 2x4's the required height, place and secure vertically, with a 2x4 across the top. Sheetrock, insulate, paint!
First you need frame out the half wall with 2X4's. If it's going to be attached to an existing wall, make sure it matches the depth of that wall (taking into account the sheet rock). When framing the wall, make sure it's plumb & level. Attach it to s solid structure (floor or existing wall) & cover in sheetrock.
Go to the lumber company and purchase enough 2x4's to make a frame for the wall. Attach the frame to the side wall and the floor. You will need 2 dry wall pieces the size of the wall, one on the front and back and you will need to decide what you want for the top of the wall.
First thing is to determine how long and high the wall is going to be. Purchase all of the materials needed. If the new wall will be attached to an existing wall, the studs need to be identified for proper mounting. Now you can start building the wall.