Ceramic tiles are painted with a glaze that hardens into glass in the kiln. This smooth surface is nonporous and impermeable, so true discoloration is rare. Before attempting to cover the stain, try removing it with bleach, vinegar and baking soda, cleansers and anything else in your cleaning arsenal. Removing the stain is the only way to restore a pristine finish -- all repair methods leave some trace, although it may not be immediately obvious to someone who doesn't know it's there.
Most home improvement stores and many appliance stores sell tiny bottles of touch-up paint that are usually used to disguise chips in appliance enamel or bathtubs, but these can also cover small stains on ceramic tile. These products paint on like nail polish and dry to a glossy shine -- use as thin a coat as possible to maintain the smoothness of the tile. The only problem with these products is that the color choices are limited -- if your tile isn't white, black or almond, you may be out of luck.
Take a spare piece of your tile to a paint store and have them use it to custom mix the exact color you need in an oil-based paint. Paint this on the same way you would the commercial products, then coat with a very thin coat of waterproof two-part epoxy. Without the epoxy, the paint would scrape right off the tile the next time you scrub, so don't skip this step. Again, this solution is only suitable for very small areas, and must be applied with finesse to prevent a lumpy mess.
Home improvement stores sell a special kind of paint called "tub and tile" paint. It comes in gallon buckets like regular paint, but it is formulated to adhere to slick surfaces like tile. These products are used to coat all the tile in the room -- you can use the same color you already have for a good sprucing-up, or opt for a different color entirely. Use these paints with a sprayer -- a brush makes it almost impossible to achieve a convincing tile-like smoothness. When done well, you won't be able to tell that the tile has been painted, although the grout lines will be painted over.