Toy bears were one of those serendipitous inventions that occured independently in two places, like the discovery of the structure of DNA, only cuter.
One source of origin was Germany early in the twentieth century. Toy manufacturer Margarete Steiff began to design and make toy bears based on her nephew's sketches of ursines in the local zoo. Her bears were fuzzy and brown, stuffed with excelsior, and had articulated shoulders and hips so their limbs could move. They had a real bear's humped back and long snout, with the familiar embroidered nose. In 1903 she displayed her bears at the Leipzig Trade Fair, and her wares so captivated an American buyer that he ordered 3000, a huge number at that time. By the end of that year the order had increased to 12,000 bears, and by 1907 the Steiff Company had sold over a million of the lovable toys. Today, rabid hordes of teddy collectors would kill to obtain a Steiff bear.
Meanwhile, in 1902 in the USA, president Theodore Roosevelt went bear hunting in Mississippi. He refused to shoot a chained orphaned cub, considering it unsportsmanlike. (Good for him.) The event was immortalized in a cartoon by Clifford Berryman of the Washington Post; it shows Roosevelt turning away from a cute bear with Mickey Mouse ears. The cartoon was seen by Rose Michtom, who along with her husband owned a toy store in Brooklyn; inspired, she would make a toy bear, put it in the window, and when it sold, would make another. Her bears had wide button eyes and soft plush fur. Her husband wrote to Roosevelt to ask permission to call the bear "Teddy", after him; permission received, they called the toys "Teddy's bear". This company's manufacture of the popular toy rocketed it to fame and fortune, first as the Ideal Toy and Novelty Company and later as the Ideal Toy Company.
America was soon swept by bear fever, just as later it would be Cabbage Patch Kids and beanie babies. Society women took them everywhere; Roosevelt was re-elected with a teddy as mascot; and the song "The Teddy Bears' Picnic" became a hit. American and German manufacturers vied to corner this lucrative market of what was now called the teddy bear, and the British manufacturer Farnell stepped in; they would be the makers of one very famous teddy, Winnie-the-Pooh. Bears were changing; boot-button eyes were replaced by glass, and excelsior stuffing by softer kapok. In the 1920s and 30s mechanical bears became popular; some of them danced and turned somersaults. The war years stopped the manufacture of German bears, though, and after the war consumers became interested in washable bears; hence bears began to be made of synthetic fibers. Although the modern mass-produced bear in insipid pastels (The Care Bears!) may have replaced the sturdy teddies of yesteryear in the hearts of today's children, teddy bears still inspire fierce loyalty and devotion amongst adult collectors, who travel the globe and spend obscene amounts of money to obtain tattered original brown bears.
I learned much of this information watching a fun documentary called "The Unnatural History of the Teddy Bear"; for written corroboration, see teddybearandfriends.com/history.html