The main tenet was an insistence that any kind of change was impossible, and so (on the usual interpretation) was any kind of plurality. Reality was one and unchanging, and the changing multiplicity of things was an illusion.
The arguments used appealed to strict logic, and were the first to do so systematically; Zeno in particular produced a number of paradoxes which plurality and change involve.
The Eleatics' main influence - especially on their contemporaries and on Plato a century later - took the form of an insistence that whatever is ultimately real must be permanent and unchanging except for moving in space, and this influence can still be felt in philosophers like Rene Descartes (1596-1650), Gottfried Wilhelm Leibniz (1646-1716) and Immanuel Kant (1724-1804).
Also see: monism
W K C Guthrie, A History of Greek Philosophy, 2 (1965)