The shower tray, or "pan," is the base of the shower, which is built out of layers of cement and other materials, and then topped with tile. A professional should build the base to ensure it's structured properly, but tiling it is something the homeowner can do with the right equipment. You'll need to rent a wet saw at your local home improvement store to cut the tiles around the drain. Remember you will be standing on these tiles while they're wet, so get a matt or flat finish that won't be slippery.
Mark two perpendicular lines across the shower floor with your pencil and level, from mid-wall to mid-wall, intersecting in the middle. Use a square to ensure the lines are exactly 90 degrees off each other. If the lines go over the drain (they usually will), continue the lines as if the drain weren't there.
Make a "dry" layout of tiles (properly spaced but without mortar between them) alongside both lines. Add additional tiles if needed to cover the whole drain.
Mark the tiles sitting on the drain with a pencil, drawing lines on the tile surface based on where the edge of the drain is, as seen through the spaces between the tiles. Move lines back 1/4 inch on the tiles as you draw them.
Cut all the drain tiles on your wet saw, cutting slowly around the curve of the line.
Apply thin-set mortar on the floor around the drain with your notched trowel. Press the cut tiles in place around the drain, leaving 1/4 inch there.
Cover the rest of the floor with mortar, using your trowel, and set in the rest of the tiles, working from the drain area outward toward the walls. Put spacers between all of them. Cut the tiles by the walls with your wet saw. Allow the tiles to set for a day and remove the spacers.
Grout the floor, spreading the grout over it with your grout trowel and forcing it into the spaces between the tiles. Don't grout the lines around the edges of the floor. Let the grout dry for three days or more.
Run lines of caulk in the open spaces around the edges of the floor. Let the caulk dry for a full day.
Wear eye protection when using your wet saw.