The simple definition
A supergiant star is an evolved massive star with extremely high luminosity. In the Morgan-Keenan system, most are class I, split into Ia, Iab, or Ib depending on luminosity.
The category overlaps with real astrophysical complexity: mass loss, pulsation, convection, and uncertain distances can shift estimates. That is why serious guides use ranges.
Size, brightness, and temperature
Red supergiants are cool at the surface but enormous, so their total output is huge. Blue supergiants are hotter and more compact, but still vastly brighter than the Sun.
Luminosity follows the Stefan-Boltzmann relationship: L = 4 pi R^2 sigma T^4. A star can become more luminous by getting larger, hotter, or both.
What makes them recognizable
Astronomers look at spectra. Supergiants have lower surface gravity than dwarf stars, which changes the width and strength of absorption lines.
They also show strong winds and atmospheric instability. Betelgeuse, Antares, Rigel, and Deneb are useful examples because each emphasizes a different part of the category.