At least that's the thesis of Joystiq's Rowan Kaiser, who tried to list the ways the franchise spawned by Richard Garriott influenced gaming in this editorial. Since you'll have probably heard about how the franchise influenced open-world gaming and morality systems many times, here's on narrative:
Narrative: In addition to being the driving force behind open-world gaming, Ultima was also at the forefront of plotting in video games, at a time when RPGs, as a genre, was pushing gaming forward. Ultima III was one of the first non-adventure games to have any kind of in-game story development and, for the first time, conversations with various non-player characters in towns became an important, necessary way to understand the game. U5 took the virtues of its immediate predecessor and attached their exploration to a story with a villain. And the series' stories culminated in the two parts of Ultima VII, which were stunningly well-developed for the time and included one of the most shocking plot twists in gaming history.
More impressive than the overarching plot was Ultima's commitment to smaller-scale conversation and character-building. While Ultima III required some basic conversations, they tended to become a single button press and reading a line or two of dialogue. Starting with Ultima IV, the series used a more complex text system, where the player would type in keywords in order to get information, starting with the ubiquitous 'name' and 'job' as introductions. This remains one of the most complex and in-depth conversation systems in RPG history. It was simplified slightly with the move to mouse-based gaming in Ultima VII, a move which laid the groundwork for the more common click-on-the-line-you-want-your-character-to-say conversation mechanic still in use.
As a final note, Ultima was one of the first games to have you control one specific character, the Avatar, which you create. The other characters already exist with pre-determined roles and personalities, and can be engaged for conversation. This has become the default mode for party-based games these days, but it was rare in that era, where you rolled an entire party yourself, or occasionally played pre-determined characters.