There have been 3 different animes (cartoons) released in the world of Dragon Quest. It began with Abel Yuusha, which was translated partially to 'Dragon Warrior' in North America and then both Dai and Roto series (which were similar to their manga encarnations) were released. Both Dai and Roto have a movie (or more) released as well. For a comprehensive list of DQ anime please consult 'diasEquip', an excel file I am constantly updating with DQ equip information.
Episodes: 43 in Japan, 13 in North America
Year 1989

Translated in English as 'Dragon Warrior' in the early 90's.

"The story is about a teenage boy named Abel and his best friend Tiala. On Tiala's 16th birthday she receives a red jewel. Unbeknownst to her, the red jewel has the power to awaken the great dragon, which legend has it can grant eternal life to one who drinks its blood. Baramos, a daemon from the ancient and evil kingdom of Estark, desires to find the stone and gain immortality." (Quoted from Wikipedia)

Episodes: 46 episodes
Year: 1991
Movies:
1991 - Dai no Daibouken
1992 - Dai no Daibouken: Stand up!! Disciples of Avan
1992 - Dai no Daibouken: Destroy!! The Reborn 6 Commanders

30 years ago the Demon Lord Hadora fell at the hands of Aban. With revenge on his mind, Hadora has returned and now seeks out Aban. Meanwhile Aban comes across Dai, an orphaned boy on a peacefull island that he trains as a warrior. Hadora tracks Aban to the island, where Dai and company lmust fight off Hadora's menace.

Movie: Roto No Monshou
Year 1996

Based on the sketchy period between the games DQ III and DQ I. The Time of Zoma had passed and the eblem of Loto was split between 2 kings who lived prosperously for many years. The king, leaving behind a pregnant queen, leaves on a mission. He obtains an evil statue and upon his return, he starts acting strangely. Arus (the hero) and his son assinate him when his true form is discovered and consequently need to escape the castle walls when the guards come looking or them. They will meet others and learn of the threat of King Dragon and devote their life to stopping the fiend.

Image courtesy of Square Enix