"Yume no Hoshi no Button Nose" centers on Button Nose, a small, anthropomorphic creature from a distant star who dreams of exploring the universe. One day, Button Nose's spacecraft malfunctions, leading to an unexpected landing on Earth. Found by a young girl named Ai, Button Nose is taken in and begins to adapt to life on Earth while forming a close bond with Ai and her friends. The series explores Button Nose's journey of discovery, blending themes of friendship, curiosity, and the wonders of the unknown.

The narrative unfolds as Button Nose learns about Earth's culture, nature, and the complexities of human emotions. Each episode introduces new challenges and adventures, often involving Button Nose's unique abilities and perspective as an outsider. The interactions between Button Nose and the human characters highlight the importance of empathy and understanding across differences. Ai, a kind and imaginative girl, serves as Button Nose's primary guide, helping the alien navigate the unfamiliar world while also learning from Button Nose's optimistic and inquisitive nature.

Supporting characters, such as Ai's classmates and family members, contribute to the story's warmth and humor. The series balances lighthearted moments with occasional emotional depth, particularly as Button Nose grapples with the longing to return home while cherishing the connections formed on Earth. The overarching plot revolves around Button Nose's efforts to repair the spacecraft and the growing realization that Earth has become a second home.

Throughout the series, Button Nose's adventures are punctuated by encounters with various Earthly phenomena, from natural wonders to everyday human experiences. These encounters serve as opportunities for growth and reflection, both for Button Nose and the human characters. The story ultimately emphasizes the value of curiosity, resilience, and the bonds that transcend worlds.