How to Make a Cheap Light Box

Buy all the parts.
The cheapest option is 1/2" plywood, but you can make the box 'classier' by using oak 'faced' wood or even those white pre-fab shelf boards.
Make sure you get the 1/4" white (not clear) acrylic sheet big enough to cover a 20" by 20" box. The thinner pieces flex too much.
The first time I made this I used giant fluorescent bulbs for the ceiling but it blew the breaker in my basement and required a lot more wiring work. For this project, just buy two under-counter short bulbs (13") that come in sets with a bulb and a power cord already attached. (~$20)

Cut all the pieces. You can use a circular saw or if you don't have one places like Lowe's will cut them for you for 25 cents a cut. It may be easier to cut the acrylic by hand with a hack saw instead of the big circular saw.
Base - 2 Pieces 20" by 9.5"
Sides - 2 Pieces 20" by 5" and 2 Pieces 19" by 4.75"
Acrylic Top - 20" by 19" (just a hair smaller so the fit isn't too tight)

Assemble the base. Place the two 'big' sides on your workbench so that they are parallel to each other and vertical. It helps to put the two short sides in between the boards to make sure they are properly spaced apart. I like to add some wood glue to the tops before I attach the base pieces for structure and to help hold it in place while I nail. Also if you have clamps big enough they come in handy here to hold the sides tightly together.

The two base pieces should be lined up with the outer edges of the sides so that there is a 1" gap in the middle. Once you are happy with the alignment, nail the base to the sides with ~1" nails (or screws).

Assemble the sides. (If you glued the base allow it to dry a bit before continuing.) With the base attached you should be able to flip the box over and push the two short sides into place. Make sure they go all the way down to the base. Nail them in place from the sides instead of the bottom this time. There should be a quarter inch difference in their height compared to the long sides.

Mount the lights. Slide the cords through the gap in the base and then follow the manufacturers instructions to mount the lights parallel to each other and the gap in the box (one on each side of the gap). Usually this just requires a couple screws. Attach the bulb to the fixture before continuing.

Put the top on and test. you should be able to just drop the acrylic sheet into place. It will slide one way and not the other because of the difference in height of the sides and should be about level with the top of the box. Plug in the cords and admire your work!