Lasse Hallstrom, born June 2nd 1946, started off as a director when he made a short documentary in high school about a group of friends who formed a rock band. Later that year, he got his first break when the documentary was shown on Swedish television. Three years later he trained to be a television producer, and in 1974 ABBA asked him to make them some promo clips (forerunners to music videos) to send to radio stations they couldn't personally visit.
http://www.youtube.com/watch?v=oVGSKVkkyhc
After doing many videos for ABBA, Hallstrom went into directing feature length films, starting with En Kille och en Tjej (A Guy and a Gal), in 1975. After this he directed ABBA:The Movie too, but his first real hit wasn't until 1985 with My Life as a Dog, where he received a nomination Best Adapted Screenplay and Best Director at the Academy Awards. His style, while not developing much through his early work as a music video director, is much more obvious in his films.
Michel Gondry, born May 8th 1963, started making films for the band he drummed in Oui Oui. After making several videos in 1992 for the band, he caught the attention of Icelandic music artist Björk, who wanted something different for her song Human Behaviour, and following the success of the video, he worked on several others with her. Many artists looking to make a statement about their style use Gondry, as he usually makes videos with a visually impressive quality, while being simple in concept or execution.
Due to the fantastical simplicity of Gondry's work, he has been quoted by some as having "redefined the entire music video genre". When you see some of his work, it is easy to understand how, as his videos have almost a magical quality to them that doesn't appear before he made his way onto the director's stage.
Recently, Gondry won an Oscar for directing Eternal Sunshine of the Spotless Mind, and since has directed The Science of Sleep and Be Kind Rewind, all films that show the dreamlike quality of his work.
http://www.youtube.com/watch?v=gLESpHrtvxs
