It certainly yields intuition to look at chain complexes of singular simplices, coming from topological spaces: Look at the nice table on page 16 of [these notes of Behrang Noohi][1] and do the exercise he suggests there (actually the whole second chapter is about your question).

This tells you exactly what is going on for chain complexes concentrated in positive degrees, as these always correspond to spaces (via the Dold-Kan theorem). For general chain complexes it would then pay off, in terms of intuition, to study the beginnings of stable homotopy theory - pages 15-23 of [Bjørn Dundas' notes][2] give a brief idea, the first chapter of [Adams' "Infinite Loop spaces"][3] is another source for this which focuses on intuition.

Finally there are abstract frameworks describing situations (i.e. categories) where a notion of homotopy exists and they capture the essence of both the algebraic and the topological examples. The currently most popular such framework are "model categories" (chain complexes are a so-called "stable" model category). There the notions of homotopy kernel and homotopy cokernel have precise meaning and you can see that mapping cones play that role in the category of chain complexes. The friendliest starting point for this is [Dwyer/Spalinski's "Homotopy theories and model categories"][4] - chain complexes are a main example there.


  [1]: http://arxiv.org/abs/0704.1009
  [2]: http://www.uib.no/People/nmabd/pp/nord051117.pdf
  [3]: http://books.google.com/books?id=e2rYkg9lGnsC&printsec=frontcover&dq=infinite+loop+spaces&source=bl&ots=2iZ-OJIAZG&sig=7NDQby5Y2uHEd9e1zEdueTMt7js&hl=en&ei=N9jATLfBKcqUOs3e3YEM&sa=X&oi=book_result&ct=result&resnum=2&ved=0CBsQ6AEwAQ#v=onepage&q&f=false
  [4]: http://hopf.math.purdue.edu/cgi-bin/generate?/Dwyer-Spalinski/theories