Note: I am not an expert on the history of the sport of jump rope, but here is what I understand about how it unfolded in the United States, at least.  

The first jump rope tournament that I am aware of began in 1960 in the little northern Wisconsin town of Bloomer (about 3,000 people).  A physical education teacher by the name of Wally Mohrman began the event to give his students an extra incentive to exercise in the long Wisconsin winter.   I did a demonstration there once in 1981, and the speed that these athletes attain with a short hemp rope is incredible.  The tournament is still run at the end of every January and attracts competitors from Wisconsin and Minnesota who try to jump as many times as possible in 10 seconds (over 60 times!). Note also: their record is 72 jumps in 10 seconds (as I recall). [Note: The Guinness book lists a person as doing 129 jumps in 10 seconds - this claim is simply bogus!!]

The American Double Dutch League was founded in 1973 by two police detective, David Walker and Ulysses Williams.  They've been holding Double Dutch tournaments ever since and draw from teams formed in schools, YMCA's, 4-H clubs, Girls Scouts, and city park and recreation departments.  Their champions were featured many times on the Johnny Carson show and next to boxers, probably gave jump rope more visibility in the U.S. than any other organization.  ADDL no longer has a web site for some reason and it appears that its founder has set up a new organization called the National Double Dutch League.

In 1976, Richard Cendali, a physical education teacher from Boulder, Colorado, formed the first modern jump rope demonstration team - "Skip-Its.”  Only two years later, Jean Barkow, another P.E. teacher in Milwaukee, Wisconsin, had the inspiration to get kids excited about an excellent aerobic exercise and help raise money for the American Heart Association by jumping rope and raising pledges.  "Jump Rope for Heart" went on to involve many millions of children raising 10's of millions of dollars in subsequent years.  The combination of Richard's pioneering work and the Jump Rope for Heart annual event led to the formation of many other demonstration jump rope teams around the country, and around the world.  Of course, it wouldn't be long before there would be tournaments where the demonstration teams could compete against one another. .  

Richard Cendali (standing in back to the far left) and the first "Skip-Its" team, ca 1976

The first "International Rope Skipping Organization" tournament was held in Colorado in 1982.  That organization would eventually merge with an offshoot, "The World Rope Skipping Federation," to become U.S.A. Jump Rope).  Other similar organizations are also present in Europe, Australia, Africa, South America, and Asia.  The first true very broad international organization, the International Rope Skipping Federation  (FISAC-IRSF) formed in the late 1990's and holds international championships every even year.  (FISAC stands for the French initials of the organization.)

Another jump rope organization called "Nawatobi" in Japan, and referred to as "Acro-Roping" in Canada, also holds competitions where athletes jump on spring boards and cross the arms in different ways while the rope turns several times around.  I'm not familiar enough with their history, however, to make more comments.

Some young past champs from the Hot Dogs USA team!

A Brief History of the Sport of     Jump Rope