Place a stud finder on the floor and mark the locations of the floor joists in the area where you want to install the register with masking tape.
Set the register on the floor in the install location between the pieces of tape. Trace a line around the register.
Drive a nail partially into the piece of wood you're about to cut out.
Attach a blade into your Dremel appropriate for the floor you are cutting. There are blades for laminate, wood and other materials.
Start the blade and begin to cut into the floor. Cut three sides. Have a partner hold the nail as you cut out the last side so that the piece of wood doesn't fall into the floor underneath. Lift out the piece of wood.
Apply the adhesive provided with the register along the underside lip of the register. Place the register over the hole and tap it in gently with a rubber mallet.