What are the different approaches to dog training?

The approaches to training dogs, including behavioral modification, are almost endless.   Which ones are wrong?  Well nothing is wrong per say, as they all have different strengths and weaknesses.   Perhaps the only bad thing you could do is to use only one approach, when more than one is necessary to your dog.  On the other hand, everyone has to begin somewhere.  
 
The more you understand what these approaches are, the more understanding you will have when working with a dog trainer or by yourself with your dog.   There is a lot to learn in dog training, which is why there is no complete definitive book on the whole scope of dog training.  There can't be as it is always training and evolving as people learn more about dogs and how they learn or change perception.

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As individuals, dogs have different perceptions, experiences, and personal preferences.   As dog owners and trainers, we are all interested in doing different things with our dogs, such as:
 

1. Companion pet dog with manners in the house, yard, car, and beyond.
2. Dog to compete in canine sporting events.
3. A dog for work such as herding, service dog, search and rescue, or protection.
4. A dog for hunting purposes.

 
Generally, these can all have trainers who would train each one with a drastically different approach for that specific activity as well as the dog's personality make up.
 
 
What things will determine the best approach to training an individual dog in a trainer's view?
 

1. Age:  A young puppy, adolescent puppy, and adult dog are all at different learning and physical growing cycles in life.   This needs to be taken into account.  A young puppy will most likely thrive in a motivational and positive approach without corrections necessarily being introduced right away.   A green adult adolescent dog may require a more disciplinary approach due to their physical strength and lack of boundaries.  An older dog may require some care and gentle handling due to their lack of coordination and frailness at that age.
2. Temperament:  A dog's temperament can be effected by various things.   Medical conditions, phobias, breed, genetics, prior experiences in life, and things unknowable all effect a dog's overall temperament.  Generally, some approaches in dog training match some temperaments better than others.  A soft submissive temperament in a dog generally will require a positive and motivational approach.  A bossy resource guarding dog may require a stricter approach with firm handling and very strict sense of boundaries.
3. Physicality:  The general physical condition and health of your dogs will make some approaches better than others.   For instance, you won't want a dog with hip problems doing any training that involves jumping or many repetitive movements.  That is just one example.  A dog with physical handicap may have joint deteriorate with a stricter approach to repetitions and physical obstacles like jumps.  An active adolescent will thrive with an active motivational approach, again generally.
4. Owner's temperament, physicality, and age:    Since this is mostly an article about the dog, I am going to gloss over this a little bit.   However, all these same factors (combined with the dog's needs) may also call for varied approaches.  As the owner is involved in training, the right approach also has to be considered for them.
5. Motivating factors:   Dogs are individuals.  They are motivated by what motivates them.   This seems like an overly simplistic statement, I know.  Simply put, some dogs like chasing games, some dogs like tug games, some dogs like food (others believe it or not do not respond to this in a motivational way during training), some dogs like herding, some dogs like hunting, and so on and so forth.  A common thread in  motivating factors is usually an activity that ties them to their owners figuratively speaking.  However, I can't force my dog to like hunting.  They will or they will not.  Dogs still have free choice in those matters.  A young puppy fearful shut down puppy may not be motivated at all by food or toy rewards for instance.   An adult dog you want to motivate with a toy (although you usually have to build this anyways in most dogs) might be found to have no interest in that after all.
6. Things that effectively prevent an unwanted behavior:  These can vary wildly dog to dog.  Some dogs will stop at a firm no.  Some dogs will take that no as an invitation to continue a game that you aren't aware that you are playing.  
7. Reason for training:  Dog sports and activities can have different approaches to training.   What technique you may use for basic obedience will probably not be used for agility training for instance.   If you are into protection sports, your goals may include obedience but not in the same way a pet owner needs obedience.   Service dog training requires a certain dog and a long work day for the dog, and that will require yet another approach to dog training.  If you are concentrating on building drive in your dog for sport like protection, then you may want to avoid firm corrections or corrections at all for a period of time.   If you are training for obedience completion, you are going to want to introduce boundaries and rules early on of some sort.
8. Safety:  Some dog sports are particularly challenging and require very good safety measures and practices.  In other words, there is much less room for error.  Some dog's have a temperament that provides a challenge to safety to themselves and others.   The approach to that dog individually or in that sport will need to be tempered with safety in mind.

 
NOTE:  THE ABOVE ARE ALL JUST SUMMARIZED GENERALITIES AND NOT A COMPLETE DISCUSSION OF ANY OF THE POINTS.
 
There are definition differences between methods of training and approaches to training.  Try not to mix up the two.  Methods of trainings usually come under a branding name like "The Koehler Method of Dog Training".  This usually means a specific trainer has taken approaches to dog training and combined them into their method and detailed training plan.  Those approaches may be subtle to some or may be obviously geared to one or more approach strongly.  However, an approach is not necessarily a detailed training plan or method.
 
Also you will see methods or approaches listed as "electronic collar" method or approach, or "chain collar" method or approach, or even "clicker" method or approach.  Those are not methods or approaches, but pieces of equipment.  Equipment does not make up a method or approach in of itself, ever.  An approach is more of an idea of what the work will be most focused on.
 
NOTE:  RELATED TO THE ABOVE LAST TWO PARAGRAPHS, THE DEFINITIONS OF THESE TERMS AS THEY RELATE TO DOG TRAINING ARE ALL OVER THE BOARD WHEN YOU READ ARTICLES.  SO PLEASE UNDERSTAND, THIS IS THE UNDERSTANDING THAT I HAVE COME AWAY WITH AFTER 13 YEARS OF DOG TRAINING AND FOLLOWING VARIOUS ARTICLES, BLOGS, BOARDS, AND SEMINARS.   ANOTHER TRAINER MAY HAVE A COMPLETELY DIFFERENT DEFINITION THAN I DO.
 
Here are some basic approaches to dog training:
 

1. Positive approach.   Some will use the incorrect term "purely positive".  Unless someone is able to train without a restraining device (a leash) or a containment device (a room or crate) or any kind of verbal correction, then it is not Purely Positive.  A so called "purely positive" approach would therefore also have to ignore the environmental and societal corrections that come as a result of not being able to restrain or contain your dog (it's a confusing topic for a pet owner)  Often times a head halter or harness type of device is used by a trainer who says they are "Purely Positive" however these devices give a correction that never ends to the dog.  Not what is intended by the phrase "Purely Positive".   Those are management and correcting equipment devices.   This approach would be discussed as one of the four quadrants of Skinner's operant conditioning work.  So a positive approach (usually they are referring to positive reinforcement), would be one that focuses of food rewards, toy rewards, and other ways to motivate the dog.  A trainer with this approach may not like using the word no, or stressing the dog ever, and so on.  ****Positive reinforcement is understood to be adding something to increase a wanted result or behavior, simply put.
2. Dominance approach:   I really don't know a successful dog trainer that works towards the approach that everything is punitive, aversive, or to be earned (everything earned) by the dog.   Usually this approach is not used (meaning using only punitive corrections or performance to earn everything) by a successful dog trainer.  Not to say parts of this approach might not dovetail with others like "nothing in life is free" or NILIF, which you will hear most dog trainers mention at one time or another.  So many things could be thought to be under this heading as parts that may be used like what kind of equipment, or alpha rolls (not recommended by me), or the way you move your body to be intimidating, and so on and so forth.   Those pieces can be found as suggested equipment or directives in thought out training plans or methods.  Some trainers would think of this as using positive punishment  only.  Often they are not thinking of the scientific studies that brought forth this name, which simply means adding something to decrease unwanted behavior or action.  Also this is thought of as training through mainly or mostly corrections. 
3. Engagement and motivational approach:  This is approach that uses active motivation to increase a dog's engagement and focus on their owner.   This is very two way for the dog trainer (or owner) and the dog.   This is a very team work approach that is deeply entrenched in the relationship between owner, dog, and dog's work that comes out of it.
4. Balanced approach:  A balanced approach to dog training recognizes that dog's need a blend of both rewards and discipline.   It is an approach that values using all four operant conditioning quadrants as their approach teach and train a dog.  Most dog trainers are using some form of this even if they lean more towards so called "purely positive" or a so called "dominance" approach.
5. Scientific approach:   This is an approach that seeks to use various scientific studies and experiments to break down their approach to dog training.   Only problem is with the way scientific studies are funded and from whom.  Also, as you know scientific knowledge can change and even change back.   It is hard to find a really sturdy scientific study or experiment with out several hundred (at least) flaws in the plan.  I am not saying it's bad to take this into account, just that if the information is flawed it will send you down the wrong path.   Also laboratories are not real life living conditions.
6. Naturalistic approach:  Some might think I am referring to the scientific approach, but I am not.  Some trainers observe what dog's naturally like to do (sniff, run, dig) and build that actively into their training approach.  They also study the anatomy of the dog and how that works to decide how to approach training a dog.  Sometimes they do call this scientific, but I am think most are referring to scientific studies they have read and based their approach on when they say "scientific.  Anatomy arguably could also be grouped into a scientific approach.
7. Traditional:  When people refer to this approach, they are referring to some earlier branded methods of dog training, like Koehler Method of Dog Training (among others).  These are usually back before the internet where the public had access usually through books and trainers who were mostly taught through these books or obedience/breed clubs.      I would say balanced approaches are geared more towards obedience exercises trained in American Kennel Club trials for obedience (possibly other popular sports in dog clubs as well).

 
***  Other approach terms you may hear are modern, carrot and stick, pack, drive, relationship, and so on.  I probably could not find or list them all.
 
As you can see, there is a lot to consider for a dog trainer or dog owner to know how to start a training plan or method.  Note most methods and plans will incorporate more than one of these, whether the trainer realizes this or not.  So a trainer leading to the positive approach may still use things that are not 100% strictly positive.  A trainer using a naturalistic approach still may have some structure that goes toward the balanced or motivational approach as well.
 
Myself?  I use a balanced, motivational, traditional, and positive leaning approaches in my method (yet to be named or fully developed) and plan (slides by individual dog on where the approach is going to be most centered on).