This is a very easy wall shelf to make. It is constructed using rabbet, not rabbit, joints. Poplar isn't the best staining wood, but if you use pre-stain conditioner it can turn out alright.
Free plans are available in PDF and SketchUp formats.
Step 1: Resaw the Wood to Thickness (if Desired)
Resaw the wood into about 1/2 inch thick for the sides and the shelf.
You can use a planer, hand plane, or sandpaper to remove the saw marks.
Step 2: Cut to Sides and Shelf to Length
Cut the sides the same length as their width.
Cut the shelf to length.
Step 3: Cut Curves on Sides
Tape the two sides together, spray glued on a cutting guide, and cut it with a bandsaw, jigsaw, or coping saw.
You can use spindle sander, sanding drum, and/or hand sanding to smooth the curve.
Step 4: Route the Rabbets
Use a straight bit in the router table to cut rabbets the thickness of the sides pieces. Make a rabbet on one long and both short sides. You can also use a specialized rabbeting it it is the right size.
Set the router table to cut to the 3/4 thickness of the backer board. Widen the rabbet on the long side of the shelf and cut the rabbets on the sides.
Dry fit the shelf and the sides and cut the back to fit.
Make 1/2 inch holes about halfway through the back for the mounting screws. Make them 16 inches apart, or whatever is the distance for wall studs in your install location.
Step 5: Glue Up
Glue and clamp the shelf assembly.
Step 6: Trim (if Desired)
If the fit on this rabbet stinks, you can cut some thin strips and use it for trim.
Glued them on and cut off the excess.
Step 7: Apply a Finish
You can use stain and spray lacquer for a finish. Make sure you use a wood conditioner if you are using a blotch prone wood.