Tarantulas, like all arthropods, must shed their exoskeletons to grow.  This act is called ecdysis or molting, and is a very important facet of a spider's life.
       The process of molting begins long before a tarantula's carapace finally splits off and a shiny "new" spider begins sliding out of its old shell.
       The tarantula will almost always refuse food when it feels a molt approaching.  Some spiderlings may continue to eat until just a few days or even the day before a molt, but this time of fasting in preparation grows longer in duration as the tarantula ages.  Adults may not eat for several weeks when getting ready to shed.  Sometimes their abdomens will appear swollen, though there has been no increase in the amount they're fed.   Also, a normally hyperactive species may appear listless and "depressed."   If the tarantula has a bald patch on its rear from flicking urticating hairs, the patch will begin to darken.  Younger spiders and spiders with bald spots due due urticating bristle loss will gain a "sheen" to the opisthosoma that indicates ecdysis is imminent.
      If a tarantula is acting sluggish, has a darkened and/or swollen abdomen, and is refusing food, it is best to make sure there are no prey items (crickets, etc.) in the container.  When a tarantula molts, it is completely vulnerable and sometimes crickets or other food items try to eat the spider or cause it unnecessary stress.  In fact, hungry crickets can and will kill a molting spider, particularly young spiderlings.
     Just prior to ecdysis, the tarantula usually spins a sort of cradle to lie in while removing its skin.  Species with urticating hair will often line this cradle with the irritating barbs to prevent predators from having easy access to the spider during a very weakened moment.
      The spider will lie in its cradle, most often on its back.  It is vitally important that the tarantula not be disturbed during this period, for shedding is a stressful activity that consumes every ounce of its energy.
        The tarantula will pump fluid pressure in its body to get the carapace to pop off first.  The opisothoma or abdomen will split along its sides, and the spider will continue to slowly, almost imperceptibly, pump fluid in its limbs to ooze the old skin off its legs.  The process can take anywhere from 15 minutes to several hours.
        Once freed from its shell, the tarantula will be like rubber.  It is very fragile indeed during this time, and needs to be left alone to continue to flex its "new" legs, to fill them with fluid so that they attain their proper shape and proportions.  Below is a photo of a young male Acanthoscurria geniculata molting:
 

     After shedding, the spider will almost glisten for a while due to the fluid that was between its old skin and the new exoskeleton.  Its fangs will be white and rubbery; the tarantula won't be able to eat until they harden.  It will sit there, defenseless, until its new exoskeleton dries.  Spiderlings may be sufficiently hardened to consume food within two days, but it may take up to a week or ten days for adults to resume eating.

    A tarantula's shed skin is a great opportunity to learn more about their anatomy.  Along with the complete exoskeleton, spiders shed their fangs and chelicarae, their throats and stomach lining, female genital organs, and the linings of the book lungs.
    With a molt, it is easy to determine the gender of a tarantula.  Females will have two little tendrils or a flap of skin protruding inward from the opening of the epigastric furrow (on the slit that runs between the bottom edge of the two white patches closest to the cephalothorax).  Those are spermathecae and are used for storing sperm.  Males do not have such organs.  While there are other methods that can be done with the live tarantula itself (examining the shape of the epigastric furrow, looking for a darker circle between the two forward booklungs on males, etc.), this method is the easiest and most doubt-free if a shedding is available.