Tae Kwon Do as it was spelled in the mid 50's by the Korean Kwan (martial art school system) leaders has separated into several large bodies over time with varying priorities and method. Despite these differences the family root cannot be denied and the original value system has persevered in common with all. It should be stated these values may be worded in slightly different fashion, but universal concepts such as Respect, Integrity and Indomitable Spirit are the core  philosophy.
It is the responsibility of each school head to ensure these values remain in focus.

Method
Tae Kwon Do is a martial art form that evolved over many decades starting in Korea. It's method is primarily kicking, although various blows by hand, head, foot, knee, and elbow may also be applied. It has four main components of practice and demonstration: Form (Poomse), Free Sparring (Kyorugi), Self-Defense (Ho Shin Sool), Breaking (Kyukpa). Tae Kwon Do has developed into both a traditional Martial art and Olympic medal sport. Olympic sport rules limit blows by closed fore-fist and feet only. No punching above the throat base is allowed, although kicks to the face are legal. Today only sparring is demonstrated at the Olympic level.

Sport
Christened in 1955 as a new martial art, Tae Kwon Do eventually developed into an Olympic medal sport. The ITF (International Taekwon-Do Federation) had considerable success in bringing it to the world through the 60's and early 70's. They currently maintain millions of members in 120+ countries worldwide. Beginning in 1972-73, Kukkiwon and the World Taekwondo Federation (WTF) became the first  (1980) Tae Kwon Do organization recognized by the International Olympic Committee (IOC).In 1987 Taekwondo was included as an official sport at the Pan Am Games and quickly became the first event to be sold out. In 1987 the first Women's World Taekwondo Championships were held. In 1988 and 1992 Taekwondo appeared at the Olympics as a demonstration sport. Taekwondo was adopted as an official medal sport of the Sydney 2000 Olympics and is a confirmed participant in the 2004 Athens and 2008 Beijing Olympic Games.  At present Olympic rules for Taekwondo are derived from the WTF, headquartered in Seoul Korea. Although WTF currently recognizes only Kukkiwon certification, events sanctioned by the BC Tae Kwon Do Association do not discriminate nor restrict based on TKD style, focusing on access for all, introducing the Olympic rules to a larger audience. This concept is a constant in Olympic games and global sports in today's world.

Martial Arts ~ A definitive description, courtesy Wikipedia ...
 (a word of caution, data at Wikipedia is not subject to verification as a condition of posting and some errors in fact are significant, please cross reference)
"Also known as fighting systems, are bodies of codified practices or traditions of unarmed and armed combat, often with the goal of developing both the character of the practitioner as well as the mindful, appropriate, controlled use of bodily force.

The martial arts, due to a century of exaggerated, exoticized portrayals in popular media (see orientalism), have been inextricably bound in the Western imagination to East Asian cultures and people, but it would be incorrect to say the martial arts are unique to Asia. Humans have always had to develop ways to defend themselves from attack, often without weapons, so it would not be correct to think that unarmed combat originated from East Asia. But what differentiates the martial arts from mere unarmed brawling is largely this codification or standardization of practices and traditions, many times in routines called forms (also called kata, kuen, tao lu, or hyung), and above all, the controlled, mindful application of force and empirical effectiveness. In this sense, boxing, fencing, archery, and wrestling can also be considered martial arts. More... "

Why is Taekwondo so popular?
Perhaps because it is so much more than self-defense skills. Joy in executing the motions, belonging and security in it's structure. It is a life-style of physical and mental training proven to develop the complete being - physical and mental, joined by an indomitable spirit. In today's world, the values taught in Martial Arts are a cherished weapon against the negative influences our children face each day. It is the individual Instructors and Students at the classroom level which have given it life above and beyond the promises the style holds.

More on the KukKiWon & World Taekwondo Federation
The Symbiosis Which Forms Olympic Style Taekwondo

The WTF or World Taekwondo Federation on the other hand serves as the current International Federation (IF) for purposes of the International Olympic Committee (IOC). As the IF for Taekwondo, the WTF regulates all aspects of sports competition for Taekwondo. It is the WTF that hosts the World Taekwondo Championships, and all other World events. It is the WTF that certifies the people who serve as International Referees and conducts International Referee training. It is the WTF who institutes the changes, if any, to the competition rules. It is also worthy to note, the WTF requires a  4th Dan to recognize one as an International Referee and only recognizes Taekwondo Dans registered at the Kukkiwon. Although the WTF has many member countries it primarily allows individuals enforcement and adjustments in their domains without interfering. Hence the inconsistency of issues such as rank for instructor (4th Dan Korea-USA-etc, 5th Dan Canada, 6th Dan Britain) Problems initiating in a member country, appealed to the international body are generally referred back to the source country, at times creating confusion and frustration.