A finite spectrum is a spectrum presented by sequences of finite homotopy types (finite CW-complexes). Precisely, they are the CW-spectra with a finite number of cells of a spectrum.
Finite spectra are the compact objects in the stable homotopy category. The full subcategory on finite spectra is equivalently the Spanier-Whitehead category on finite CW complexes (e.g. Schwede 12, chapter II, section 7).
Many of the basic constructions and theorems in chromatic homotopy theory apply to finite p-local spectra, such as
Neil Strickland, Section 2.2 of An introduction to the category of spectra (pdf)
Stefan Schwede, chapter II, section 7 of Symmetric spectra, 2012 (pdf)
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