High-temperature superconductivity is a historical name for superconductivity above earlier metallic benchmarks. The original Bednorz-Müller experiment operated at tens of kelvins, requiring deep cooling; cuprate superconductors opened a new materials family.[ref]
Perovskite-type oxides: The new approach to high-Tc superconductivity by Johannes Georg Bednorz and Karl Alexander Müller. Part 1, p.426, historical comparison
A higher superconducting transition temperature enlarges the available operating range, but useful transport also needs a sufficient superconducting critical current at the operating temperature and magnetic field. In a ceramic, connections between grains can limit the whole specimen even when individual grains support superconductivity. This connects the materials problem to superconducting weak links and the Josephson effect. The parallel-path model of a superconducting ceramic below isolates why connectivity matters.
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As of 2020, basically means "liquid nitrogen temperature", which is much cheaper than liquid helium.
The dream of course being room temperature and pressure superconductor.
Timeline of superconductivity from 1900 to 2015
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