Caulk is used to seal the joint where your bathtub meets the ceramic tile. Grout keeps water from working its way between the tiles.
There's a lot of expansion and contraction along the bathtub joint, and it's not easy to keep it watertight. The problem is worse when the joint is filled with the same grout used between tiles. Grout won't flex; instead it cracks and admits water into the drywall or plaster behind the bathtub joint.
To repair a bathtub joint, it's usually best to use a 100% silicone adhesive caulk, which comes in tubes or cartridges and is available at any hardware store or home center. Look for ones called tub caulk, or tub and tile sealer. It stays flexible for years and resists cracking despite expansion and contraction. Most varieties are off-white to match standard tile grout.
Pick out all the old caulk (or grout) from the joint. Use a thin screwdriver that's slightly narrower than the joint and a razor blade or a utility knife. Clean off all debris so that the new caulk will adhere properly. Scrub with bathroom cleaner; rinse thoroughly with a sponge and dry well with cloth.
If you are using a caulk gun, cut the spout of the caulk cartridge so that it will produce a bead of caulk large enough to fill the crack. Mount the cartridge into the gun and squeeze the gun's trigger while moving the spout forward. This will force the caulk into the crack and form an even bead the length of the crack. If using a tube of caulk, you'll also need to cut the nozzle of the tube to produce a proper-sized bead to fill the joint. Squeeze the tube, pushing the caulk ahead of the spout. Smooth the bead with your finger. (Wear smooth-textured gloves.)
Sloppy caulking is a sure way to spoil the appearance of the tub joint. For professional results, take a tip from painters and reach for masking tape. Position the tape along both sides of your planned line of caulk so that your finger will plow the excess caulk onto the tape when you smooth the bead. Also, before smoothing the bead of caulk, wet your gloved finger with dishwashing liquid so that it moves easily along the caulk.
After you smooth down the caulk, carefully lift away the tape. You'll get a caulk line with crisp, straight edges--and a handful of goopy tape. Let the caulking dry overnight before using the bath. After the caulk has dried, use a razor blade to carefully remove any excess along the caulk line.
In contrast to caulk, grout is a type of water-based cement similar in composition to the mortar you see between bricks. Rather than acting as an adhesive, though, grout is a barrier to keep water and dirt from the spaces between tiles.
If some of the grout has fallen out, carefully scrape the remaining grout free with the tip of an old screwdriver or a lever-type beverage opener. Be extra careful not to chip the edges of the surrounding tiles.
Wearing rubber gloves, use your fingers to press grout into the joints between the tiles and smooth it. Wipe off excess grout with a damp sponge. After the joints dry to the touch, use a dry cloth to burnish off the haze that forms on the tile surface.