In terms of presentations this means:
for a fibration in the ordinary model structure on topological spaces or in the model structure on simplicial sets, and for the ordinary fiber of topological spaces or simplicial sets, respectively, we have such a long exact sequence.
For background and details see fibration sequence.
Given a tower of homotopy fibers such as a Whitehead tower or Adams resolution, the long exact sequences of homotopy groups for each stage combine to yield an exact couple. The corresponding spectral sequence is the Adams spectral sequence.