There're worldwide, national, regional
and breed specific organizations

and AFAIK there're mainly 4 organisations worlwide
which are covering about 100 countries and all 5 continents:
Fédération Cynologique Internationale
The Kennel Club
American Kennel Club
Australian National Kennel Council Limited

The largest and world wide acting all breeds organization is the Fédération Cynologique Internationale (FCI). It includes 86 members and contract partners (one member per country) that each issue their own pedigrees and train their own judges. The FCI makes sure that the pedigrees and judges are mutually recognized by all the FCI members.
The FCI recognises 343 breeds, each of them is the 'property' of a specific country. The 'owner' countries of the breeds (i.e. for the Maltese breed it's Italy) write the standard of these breeds (description of the ideal type of the breed), in co-operation with the Standards and Scientific Commissions of the FCI, and the translation and updating are carried out by the FCI. These standards are in fact the reference on which the judges base themselves when judging in shows held in the FCI member countries; they are THE reference assisting the breeders in their attempt to produce top-quality dogs.
Every member country conducts national as well as international shows (conformation shows). International results are sent to the FCI office where they are processed. When a dog has been awarded a certain number of awards (CACIB), it is eligible to receive the title of an International Beauty. The title is confirmed by the FCI.
In addition, via the national canine organisation and the FCI, every breeder can ask for international registration of his/her kennel name.

The UK's largest organisation dedicated to the health and welfare of dogs is the Kennel Club. Originally founded in 1873, the Kennel Club's sole consideration was to regulate the conduct of dog shows and trials and then later to register dogs to avoid the duplication of a name in the stud book. Whilst these functions remain, the Kennel Club has developed in ways that could not possibly have been imagined when it was established in Victorian times.
They have a voluntary register for all dogs, whether cross breed or pedigree, and they run an Accredited Breeder scheme whose members have agreed to follow basic responsible breeding practices for breeding puppies. The Kennel Club runs Crufts, the world's largest dog show and licenses over 4,000 dog shows and trials each year. It is also at the forefront of the development of many canine health initiatives and invests in education programmes and other campaigns to protect and promote the general wellbeing of dogs.
The Kennel Club is concerned with anti-dog legislation, promotes the place of dogs in society and actively represents the interests of responsible dog owners, whether Kennel Club registered or not.

On September 17, 1884, a group of twelve dedicated sportsmen, responding to a "meeting call" from Messrs. J. M. Taylor and Elliot Smith, met in the rooms of the Philadelphia Kennel Club in that City. Each member of the group was a representative or "delegate" from a dog club that had, in the recent past, held a benched dog show or had run field trials. This new "Club of Clubs" was, in fact, The American Kennel Club.
In 1998, as more events type were being made available to fanciers, Agility, Obedience, Tracking and Canine Good Citizen became a separate division within the AKC. The remaining Performance Division consists of Field Trials, Hunting Tests, Lure Coursing, Herding, Earthdog and Coonhound events. Conformation continued to benefit from a steady growth on participation. In that year alone, there were almost 2 million dogs competing in over 15,000 member, licensed and sanctioned events.
Also in that year, AKC registered more than 1.2 million dogs and 555,000 litters. DNA rapidly gained acceptance not only by fanciers who embraced the new technology, but also by AKC who saw its promise as a tool to ensure the integrity of the kennel club's registry.

There can be no doubt that the Australian National Kennel Council Limited has developed co-operation between the 8 Controlling Bodies and made a large contribution towards improving dog showing, breeding and other canine activities in Australia. The ANKC Ltd will now continue this work.

A list of all known clubs also regional ones to search for per country you'll find here. And keep in mind that all above mentioned orgs don't have the same priorities. Most of them have only (voluntary) registering dogs and holding dog shows as their main task and others i.e. FCI set up the standard and breeding rules too.