Normally, you fill the space between hard tiles (like ceramic or marble) with grout, a form of cement. However, for the spaces where the tile runs along inside corners or meets other materials, you'll generally want to caulk instead. This is especially true for the perimeter around tubs, shower floor and the bottom of kitchen backsplashes, where moisture issues require the added protection of caulk.
Tile the space around the tub or shower walls, mortaring the tiles in place. Put spacers between the tiles as you hang them and between the tiles and the adjacent tub or shower floor.
Grout the tiled area with a grout float, pressing the grout into the spaces and wiping up the excess grout with a damp sponge.
Wet and wring a washcloth. Run the tip of the washcloth all along the line between the tiles and the tub or shower floor, digging out all the grout from that line. Re-wipe the tiles as needed with the damp sponge to make sure the grout in the rest of the area is smooth.
Let the grout set for a day.
Use a utility knife to cut about 1/4 inch from the tip of a caulk tube, at a 45-degree angle. Load the tube into your caulk gun. Stick a long nail into the hole in the tube to break the inner seal.
Place the tip of the caulk tube on one end of the open space between the tile and the tub or shower floor. Squeeze the trigger until the caulk comes out the tip. Drag the gun backward, slowly, along the line, filling it with a smooth line of caulk. Caulk the whole span.
Wet your thumb, and run it gently along the new caulk line, flattening and sealing it. Repeat for each open line. Let the caulk set for 48 hours.