It is frustrating, as someone who only wants to help folks with their pets, to receive panicked messages with blurry photos of injured or sick geckos and a "okay what should I do?" attitude. My answer will always be, "I'm just someone on the internet - please take your animal to the vet." The following section is just a selection of things that can potentially go wrong with cresteds, and possibly how to help avoid them, but it is not to be taken as a replacement for proper veterinary care.
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 body, tail, or 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 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. At the site of the autotomy, small flaps of skin immediately begin folding in to close the area off; the rest generally heals quickly and should 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 down to 50-60% during the day to help the gecko shed. 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 room temperature - not warm or hot - water, as water that feels warm to us may be enough to scald or stress this species). 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 or clean pencil eraser.
Dehydration is generally a symptom of something larger, but it can be helped until you can get your animal to a vet. If its dorsal skin is gently rubbed or pinched, the skin will stay in peaks like a small mountain range even if the gecko is moving. In extreme cases of dehydration, the gecko's eyes may also appear sunken even when awake, and its tail may appear zig-zagged. 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 lightly 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 - again, using the bath method mentioned above. Even if your gecko looks dehydrated, never attempt to force-feed liquids without the help of a veterinarian, as forcing liquids into an animal can cause stress, or worse, aspiration of the fluid.
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 by many 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. Most specimens spotted in the wild tend to be tailless by the time they reach adulthood, and many of our captive specimens never drop their tails because they do not feel threatened - the weakness in the base of the tail may just be a sign that this species was not meant to keep its tail into adulthood, especially with the nice fattening diets we can offer them adding weight to the tail in general. I have successfully bred animals of both sexes with floppy tail and it has not hindered their ability to reproduce. If floppy tail concerns you, 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 crested'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 "puffy" 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.
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. Juvenile geckos also do not seem to often have reserves, perhaps because their calcium goes into the growing bones. 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 and other parasites, 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. It is helpful to bring a stool sample in with almost any visit to the vet.