The name is usually taken as "sitting down near", the posture of a pupil receiving teaching not meant for everyone. Each Upanishad is formally attached to one of the four Vedas and to a particular branch of its transmission, which is how such varied material came to sit inside a single revelation.
Thirteen are conventionally called principal or mukhya: Brihadaranyaka, Chandogya, Taittiriya, Aitareya, Kaushitaki, Kena, Katha, Isha, Mundaka, Prashna, Mandukya, Shvetashvatara and Maitri. They divide by form and age. The oldest are prose, and it shows, the Brihadaranyaka and the Chandogya are sprawling, argumentative, full of named disputants, court debates and men losing arguments in public. The verse Upanishads, Katha, Isha, Shvetashvatara and Mundaka among them, are later, tighter and more quotable, and they carry theistic material the prose texts lack: the Shvetashvatara in particular reads Rudra-Shiva as the supreme being.
Beyond the thirteen the tradition kept producing them. The Muktika, itself the last item on its own list, sets the number at 108 in a dialogue between Rama and Hanuman, and the manuscript traditions of that list are not identical. The later members are sectarian or specialist, Yoga Upanishads, Sannyasa Upanishads, Shaiva and Vaishnava and Shakta texts, and some are very late indeed. The Allah Upanishad, an Islamic-inflected text, appears to date from the Mughal period. The label was, in practice, a way of claiming Vedic authority for new teaching, which is why the count is open-ended and why the principal thirteen are treated separately.