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 (the "butt") will be clean. Its waste will generally have a whitish or clear liquid section (urates) with a more solid "pellet" of brown stool (feces). Its tail will be straight without sharp curves or kinks when at rest. Its skin will be free of mites and open wounds, and it will ideally not have random pieces of shed stuck on its toes.
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. You may want to start your search for an exotics veterinarian on local Facebook groups or other social media connected to your area.
If you are introducing a new gecko to a previously-started collection, a period of quarantine is essential. Quarantine should be, at absolute minimum, sixty days, though longer is very highly recommended. I tend to shoot for 2-3 months of quarantine. 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 including under the nail. If possible, the new gecko should be kept in a separate room and fed after you take care of all of your other animals and use disposable food and water dishes. During quarantine, you can monitor the new gecko for health issues and watch to make sure it is feeding normally and leaving healthy droppings.
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, just keep an eye on the tail drop area and make sure signs of infection do not appear. Please do not put medications on the autotomy wound unless you have consulted a veterinarian - many lotions or balms made for human sores contain medications that could harm your gecko. These wounds generally heal quickly and do not need aftercare.
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 around 90% at least at night when you are misting the enclosure, and should be allowed to dry out a bit 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 fifteen minutes (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. If your home is on the dry side, you may need to mist again in the morning. In cases of extreme dehydration (possibly after shipping a gecko in warm weather), some keepers have recommended using unflavored Pedialyte (electrolyte drink for children) as a rehydrator. Even if your gecko looks dehydrated, never attempt to force-feed liquids without the help of a veterinarian, as you may choke your gecko.
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"). 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, or perhaps a weakness in the genetic line compounded with sleeping face-down. 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 indicator of dehydration, or 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. I recommend that a gecko which has suffered from - or is currently suffering from - MBD not be bred, especially in the case of the female.
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. Please be aware that continually manhandling the gecko's mouth open can cause the animal a lot of stress. You might check the calcium levels once or twice a breeding season if you are seeing an issue with egg calcification, but it isn't necessary to do so often.
There are a 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. Animals with E. invadens generally go downhill quite quickly so please speak to a vet if you see these symptoms.
Crested geckos may also suffer from pinworms, which can generally manifest in weight loss, lack of growth, and runny stool, as well as occasional worms visible in the stool. Again, it is suggested you speak to a vet if you are seeing these symptoms.