Yu-Gi-Oh! Duel Monsters refers to the title of the second Yu-Gi-Oh series, produced by NAS, rather than Toei, the makers of the first Yu-Gi-Oh! series and the first Yu-Gi-Oh! movie. Because the series subtitled "Duel Monsters" was much more popular than what Toei produced, both in North America and in Japan, it functioned as the basis for the TCG and the OCG. Outside of Japan, the series does not have the subtitle "Duel Monsters," which leads to some confusion regarding whether it has any connection to the first/original series.
Because the two series were produced by entirely different studios, the two have no connection beyond the characters and some of the story being presented. Toei's version more closely resembled the early arcs of the manga, while Yu-Gi-Oh! Duel Monsters took off from approximately volume 7 of the manga, when Yuugi Mutou and his friends embarked for Duelist Kingdom. In addition, though both series feature "filler" episodes that were not part of the original manga's storyline, the Yu-Gi-Oh! Duel Monsters series features a great deal more of these "filler" episodes, including the Virtual World arc and the Orichalchos Saga, sometimes also called DOMA (or DOOM).
The movie produced originally for American audiences, Yu-Gi-Oh! The Movie (called "Yu-Gi-Oh! The Movie: Pyramid of Light" in Japan) is based off the Duel Monsters series. Likewise, the sequel/spinoff known as "Yi-Gi-Oh! GX" is properly subtitled "Yu-Gi-Oh! Duel Monsters GX" in Japan, since it is based off the storyline created in the second series of the anime.