It's sad but true, hamsters do get sick. Here are some common ailments, symptoms, and cures.

Wet Tail
symptoms - diarrhoea, stiffness, inactivity, strong musky smell, wetness around the tail, loss of appetite.
cure - take your hamster to the vet ASAP. Meanwhile, feed the hammie high-protein foods.

Colds
symptoms - sneezing, runny nose, standard cold symptoms.
cure - keep hammie warm and give him lots of vitamins.

Abscesses
symptoms - puss, infected wound.
cure - See the vet.

Cancer
symptoms - tumors.
Cure - Believe it or not, hamsters can get cancer too. It usually occurs in older hamsters but there's not cure for it :'( (luckily it's not too common)

Teeth Problems
symptoms - overgrown teeth.
cure - Always make sure your hamster has something hard to knaw on because its teeth will never stop growing all throught its life. Chew sticks and walnuts (in the shell) are good suggestions (I use these). See the vet and they can clip the teeth if everything else fails.

Overgrown Nails
symptoms - long curling nails.
cure - This is very rare and you shouldn't have any problems with overgrown nails. If you do, though, bring the hamster to the vet! NEVER try to clip your hamsters nails yourself or you may kill it if you hit a blood vessel. If they're just mildly long, let your hamster run on fine sandpaper to file them.

Dehydration
symptoms - very thirsty.
cure - always make sure their water bottle is full and working correctly. Run your finger over the spout to make sure water is coming out. To see if your hamster is dehydrated, pinch the skin (not hard) and if it goes back quickly it's fine, but if it stays, your hammie is seriously dehydrated.

Hibernation
symptoms - contstant sleeping, seems dead sometimes.
cure - Your hamster will hibernate if kept at a temperature lower than 50°F. Gradually warm the temperature in the room and the hamster should wake in a day or two.

Pregnancy
symptoms - large stomache, seems overweight even though it has a wheel.
cure - This isn't an illness, it's an adorable gift :) Hamsters are great mothers and will need little help from you so just make sure they have plenty of food, water, clean bedding, and leave the babies alone once they're born!! (see breeding)

Bad Odor
symptoms - your hamster stank!
cure - Hamsters will smell as bad as their cage. Clean the cage weekly and try putting more bedding in. Make sure to thoroughly clean places where he urinates as this is more than half the smell. Never bath a hamster! It can die.