As with any living creature, it's important to know ahead of time what to look for in a healthy crested gecko.
A gecko in optimal health will have clear, bright eyes which are equal in size. Its vent will be clean.
Its waste will generally have a whitish or clear liquid section with a more solid "pellet" of brown stool.
Its tail will be straight without sharp curves or kinks when at rest. Its skin will be free of mites, and it will not have random
pieces of shed stuck on its body.
It is important that you research veterinarians in your area before committing to a reptile pet. Many vets who are excellent with your dog or cat may never have seen a crested gecko, much less treated one, so it's important to look for someone who knows their way around exotic animals.
If you are introducing a new gecko to a previously-started collection, it's a good idea to quarantine it for at the very least thirty days, though longer is recommended. In this time, you should take precaution to not handle your new gecko before you handle your original stock; if you do, wash your hands thoroughly. If possible, the new gecko should be kept in a separate room. During quarantine, you can monitor the new gecko for health issues.
A small scale which measures to the nearest tenth of a gram is a good investment toward the health of your gecko; when used once or twice a month, this will help monitor your gecko's growth and help identify any sudden weight loss, which can be a sign of a larger problem.
Things to watch out for: If your gecko has dropped its tail, there may be a small amount of blood present. You can help your gecko heal well from a tail drop by keeping its enclosure immaculate if you are keeping him on paper towels. In the case of using a naturalistic substrate, it may be worth it to remove a gecko which has recently dropped its tail to a paper-towel-lined enclosure for a week or two, to give the tail "nub" time to heal cleanly.
An incomplete shed, or shed skin which was not removed by the gecko, can cause constriction on digits and
the tail. If you find that your gecko is having issues with shedding, you may be keeping the enclosure either
too dry, or too moist. Adjust the humidity accordingly. Humidity should be moderate at least at night when
you are misting the enclosure, and should be allowed to dry out for the most part during the day. If your gecko
is having trouble removing skin that should be shed, you may put it into a small Tupperware-type container
with a very moist paper towel (use lukewarm, but not hot, water). With a few small holes in the container for
breathing, you can leave the gecko inside for about a half hour (monitor it to make sure it's okay inside).
When you take it out, you can attempt to gently rub away any stuck shed with your fingertips, or use a
cotton swab.
If a gecko is dehydrated, if its dorsal (back) skin is gently rubbed or pinched, the skin will stay in peaks like a small mountain range. It is essential that your gecko be properly hydrated. Always provide clean water, and mist the gecko once a night. In cases of extreme dehydration (possibly after shipping a gecko in warm weather), some keepers have recommended using Pedialyte (orally-taken electrolyte drink for children) as a rehydrator.
A tail which is bent near the vent to one side, or up over the gecko's body, is called "floppy tail syndrome"
(sometimes "FTS" on various internet forums). Formerly
thought to be caused by a diet lacking in calcium, it is now generally believed that floppy tail is brought on
instead by geckos sleeping face-down on branches or enclosure surfaces so their tail will flop over their heads.
While this is now not thought to be an indication of poor health, it can lead to hip problems. If you find your
gecko is sleeping upside-down often, try lifting it off of the surface it's on and setting it on a more horizontal
surface. Also, always make sure your gecko has plenty of items in its enclosure for its tail to wrap around as
it sleeps. 
A tail which is wavy or zig-zagged can be an early indicator of a diet lacking in calcium. Proper calcium is
crucial for the crestie's health, and if left unchecked, a calcium-poor diet can lead to metabolic bone disease
(or MBD). Signs of MBD include swollen legs and a soft or "mushy" looking jaw, as well as an inability to
adhere to surfaces (though this last may also be signs of an impending shed). This disease can lead to
fragile or broken bones, so it is incredibly important to make sure your gecko is getting a proper diet. This is another reason
I highly recommend Repashy's food mixes - especially for beginners, it takes the guesswork out of balancing
a gecko's nutritional needs. It is recommended that a gecko with MBD not be bred. 
You can attempt to check your gecko's calcium stores by rubbing the sides of its mouth with your index
finger and thumb. In some cases, this will encourage the gecko to open its mouth. You can prevent the gecko
from closing its mouth entirely by using a cotton swab with the cotton removed from both ends. The calcium
sacs appear as two tiny white "bags" in the back of the gecko's mouth. Sometimes, in adult male geckos, the
calcium sacs may appear to be small. This is apparently normal, as males do not need as much calcium in reserve
as females do; the calcium goes into creating the shells of eggs.
If one eye suddenly grows larger than the other, it could be an indication of an infection. Take your gecko to
the vet.
There are few currently-known parasites or diseases specific to crested geckos, but if you find your gecko is losing weight and acting much more sluggish than usual, it's worth your time to get it tested for Entamoeba invadens. I know little about this disease personally, so please read up on it at ciliatus.com here (warning, graphic dissection pictures).
