Th "slime" is mucous from irritated bowels. We usually see this when a dog has parasites such as worms or a protozoan infection such as coccidia or giardia.
The most common worms found in dogs are:
Roundworms, Hookworms, Tapeworms, and Whipworms and Heartworms
In some cases these worms are passed from the mother to the puppy, in other cases such as tapeworms, they are contracted by other methods such as ingesting an infected flea. Most over the counter medicine will only kill one or two types of worms, so your best bet is to take a fecal sample to your vet to determine the type of worms your dog may have and then decide on treatment. Dogs should be checked for worms periodically even if they appear healthy. This is so if your pet does become ill, you are not fighting a worm infestation along with a more serious problem.
For additional information about worms, see this site
Common dewormers are Panacur which is a prescription product so you need to obtain it from your Vet and it works on roundworms and whipworms. Wormers labeled as Pyrantel Pamoate are the best over the counter medication for roundworms. Drontal plus is also a prescrition product and is used for the treatment of Tapeworms. Revolution and Interceptor as well as other heartworm medications also control whipworms and roundworms. Another reason to keep your dog on Heartworm medication.
Information on coccidia
Symptoms are diarrhea which could be mild or severe depending on how infected they are. If severe there may be blood and mucous along with vomiting, loss of appetite leading to dehydration.
Information on giardia
Symptoms of giardia are diarrhea with possible terrible smell, mucousy stool and sometimes bouts of vomiting.
I hope you find this information helpful.