Moulting
The moult is the most amazing event. There are many subtle signs that show a moult is iminent, but it can take some time to read the signs accurately.

One of the best and most obvious is when your pet has a bald spot on the abdomen.
The tarantulas skin is pale in colour, but near a moult it will start to turn black, due to the new bristles growing on the new skin under the old, and showing through.
Another good sign is your pet not wanting to eat. Don't worry, just make sure you do not leave any prey items in the enclosure for more than 24 hours, thus reducing the risk of the prey nibbling on the spider and possibly killing it.
Fresh water should always be available.
Many other signs depend on an intimate knowledge of your spider. Some will disappear into their burrows and seal the entrance up, and you may not see them for a couple of months. Again do not worry, they will emerge when they are done.
Your spider is very vulnerable just before and after the moult. Before they will have a muffled contact with the world and reaction times will be slower, and after they need the chance to stretch and harden the new skin.
When your spider turns on to it's back, the moult has started. Do not touch you spider during this time as it could cause irreparable damage.
Your pet is NOT dying!

It may take some time for your spider to move. Several hours is normal. Then slowly the old skin will split and the spider will make his way out of the old skin, using almost imperceptable rythmic undulations of the legs.
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Young specimens may only take a couple of hours to finish the whole process, but as your pet matures, the process will take longer.

After this the spider is understandably exhausted, and in bigger specimens you may be able to see the heart beating hard and fast.

The moulted skin may still be attached in places, especially the book lungs, which are generally the last to separate due to the complex design, so please do not try to take the shed out until the spider moves away from the old skin.
A newly moulted spider will be unable to defend itself, as it's fangs will be soft and white to begin with, and will change to pink, red and then back to hard black. This may take a couple of weeks.
The spider will find it hard to run away as the legs will be weak  and some what longer, it will take a while for the spider to gain full control again.

Do not handle your spider during this period, when the tarantula starts eating again, then you can safely assume the routine you had before, unless the spider is in feeding frenzy following the moult and then it makes sense to wait until it's appetite slacks off.

Factors such as food availability, heat and humidity can affect how often your pet moults, temperatures below 75 degrees will invariably slow down the metabolism so growth will be slower, and this could "arguably" have a detrimental affect on the eventual adult size of your tarantula.