Although Englishman John Crew Wood patented a form of laminated safety glass with a layer of tree resin in 1905, the real credit for the discovery belongs to France in 1903. Edouard Benedictus (painter, composer, and chemist; 1878-1930) accidentally knocked over a glass flask while working in his lab, and was surprised to discover that it failed to shatter. A brief discussion with his assistant revealed that the flask had contained a small amount of liquid plastic (celluloid), which had evaporated, leaving a transparent layer of plastic on the inside of the flask. Because it was invisible, the assistant assumed the flask was clean and put it away without washing it. When the flask hit the floor, the layer of plastic held the shards together, preventing it from shattering. After numerous experiments, he received a patent for his invention in 1909.
As automobiles were relatively new in the early 20th century, Benedictus saw an application for this discovery in windshields. Many injuries from car accidents at the time were compounded by flying glass from shattered windshields, and his safety glass could save lives. Unfortunately, the developing automobile industry had no incentive to spend the extra money on his invention. If one car was safer but more expensive, the average consumer would probably buy the less safe but cheaper model. It wasn't until 1937 that safety glass became mandatory for car windshields, and few manufacturers included the option before then. It is perhaps interesting to note that windshield wipers were also made mandatory in this year.
Benedictus also developed the idea of using a gelatin layer to adhere the plastic layer to the glass layer in 1910, creating the first three layer safety glass, called Triplex. This version of safety glass was the first to see real widespread use – during World War I. The military quickly saw the advantages of using glass that would not shatter in such delicate and high-risk applications as airplane windshields and gas masks.
Early types of safety glass were made with celluloid, but celluloid becomes brittle and discolored with age. Modern safety glass uses vinyl, specifically polyvinyl butyral, or PVB. Today, the Dupont corporation holds the Benedictus awards in honor of his discovery, which recognizes "the innovative and outstanding use of laminated glass in architectural projects worldwide".
Some clarifications: Sources vary on some minor but significant points. As far as I can tell, Edouard Benedictus first discovered the laminating process for safety glass in 1903, but John Crew Wood beat him to the patent office with his tree resin version in 1905. Benedictus patented his first celluloid version in 1909, and then added a gelatin layer to patent Triplex in 1910. No manufacturers seemed interested in using safety glass in windshields before 1920, although it is unclear who was the first to do so before it became mandatory in 1937.