Here is a quick guide to the different types of training equipment, how to identify them, and what they are used for.
Flat Leash & Buckle Collar
This is the classic combo that you think of when you think of a dog being walked. It consists of a cotton, nylon, or other material collar that clips together with a buckle of some sort and a 4-6ft flat leash that can also vary in material.
These collars were created for dogs whose head is just as narrow as there neck, allowing normal collars to slip off of them very easily. These also have an advantage not choking a dog when properly fitted. A martingale collar can only tighten up to a certain point, and from their will remain the same size.
Slip leads are often used in conformation when showing dogs, as well as in training because they are easy to slip on and off the dog. When used correctly, they give the handler a great amount of control without choking the dog.
These harnesses are designed for dogs who have a strong opposition reflex. In simpler terms, they are for dogs who love to pull their owners down the sidewalk using the force of their entire body. These harnesses allow the owner to redirect he dog's chest back towards them when the dog begins to pull, and can be extremely useful when dealing with strong dogs who have poor leash skills.
These are normal harnesses you can buy at any pet store. Unlike the front clip, they clip on the back of the harness, and allow the dog to pull with the entire strength of their body if they do not have good leash skills. These harnesses can be helpful for dogs who cannot wear a normal collar due to medical conditions in their trachea, and can also be great for outdoor activities such as hiking where dogs need more freedom and pulling isn't as much of a problem.
Like front clip harnesses, head halters are designed for dogs who pull on the leash. It can also be very useful for dogs who get very distracted during walks, even when their isn't a lot going on. They work by redirecting your dogs snout back in your direction, which in turn redirect their neck and body towards you. I personally use head halters for both my dogs and love them. I use the Walk n' Train Head Halter -- I like the safety clip it has that clips to your dogs collar in case the halter slips off.
Just because a dog is wearing a muzzle does not mean that the dog is aggressive. Dogs where muzzles for several reasons. These include: keeping the dog from eating foreign objects on the ground, the dog is working on anxiety around strangers and other dogs, they could be practicing muzzle skills for their next vet visit, or they just prefer space and muzzles keep people away.
Prong Collar/Choke Collars
As a trainer who uses positive reinforcement, I do not condone the use of choke chains or prong collars, however it is important to know what they look like and what they should be used for. These types of collars should be used to give a dog a quick correction if he shows an inappropriate behavior. They should not be used to choke out a dog or instill maximum amounts of pain to force a dog into obedience.