High temperature superconductivity was discovered in 1986 by Bednorz and Mueller at IBM Zurich. These are copper-oxide based ceramics with a "layered perovskite" crystal structure, as shown below. The cuprates, as these are called, can be made with a varying number of CuO planes per unit cell, which dramatically affects their superconducting properties. It is clear that the superconductivity originates in these sheets, but no one yet knows how.
The mechanism for the superconductivity as well as many of the details of the electronic structure are still unknown and is intensively studied around the world. The following figure shows the timeline for discovery of superconducting compounds.
In addition to the superconductivity, many people believe that the "normal state" (at temperatures above Tc where the materials no longer superconduct) is even more unusual and exotic than the superconducting state. In this view the "correlations" between electrons are so strong that the materials behave like very unusual metals which are not yet understood. Understanding these unusual electronic correlation effects, as well as of course the superconductivity, is a critical goal for this field of research.