If you're picturing a board with 121 holes, the answer isn't Twister—it's Chinese Checkers.
The iconic Twister mat (Milton Bradley, 1966) features a 6 × 11 grid of colored circles—that's 66 circles, not 121. In fact, a standard Twister mat has only 24 circles (4 colors × 6 rows).
Correction: The "121 holes" description perfectly matches Chinese Checkers. Its star-shaped board contains 121 holes (10 per point × 6 points + 61 center holes = 121). Each player starts with 10 pegs in one point of the star.
Therefore: The board game with 121 holes is Chinese Checkers.