Before attempting to fix the wall with the brown stain, find and fix the source of the water leak causing the brown stain. If you simply attempt to cover the stain without addressing the reason why it's happening, the brown stain will return. Once you have addressed the cause, you have several options for fixing the stain, one of which involves bleach and the other involving painting.
Cleaning with Bleach
Mix 1 cup of bleach with 1 gallon of water.
Scrub the stain with the mixture and a scrubbing sponge. Don't be too aggressive, but do apply enough force to work the stain out. Rinse and wring out the sponge frequently. You want a clean mixture on the sponge, but you don't want to soak the wall with water or remove paint. Wear rubber gloves to protect your hands from the bleach. Start at the top of the stain and work your way down.
Allow the cleaned area to dry thoroughly. Use fans and a dehumidifier if necessary.
Assess whether or not you were able to remove the stain. If not, you'll need to paint the area.
Ensure the stained area is completely dry. To test it, try this technique. Cut a shallow X in the center of the water-damaged area. Press duct tape over the X and pull it away quickly. If the surface is dry, the X will remain in place.
Tape off any areas you don't want painted with painter's tape and lay down a drop cloth to cover the floor.
Scrape the area clean of any loose material. Sand with medium-grit sandpaper if necessary.
Seal the stain by applying a coat of stain-blocking primer.
Apply a topcoat of paint that matches the wall. If an exact match isn't available, you may need to repaint the entire wall or room.