I know that this definitely have some sort of reference out there, but I did not find any wikipidea page for it or any introductory Mathematical article about it . I just want definition and concrete examples of translation surfaces. For example, if I take four copies of equilateral triangle and glue them together side by side to form a tetrahedron, how do I put a Riemann surface structure on it ? I want some detailed explanations etc. Could you cite any reference(s) ?

There is a huge literature, and I'm not sure exactly what you are looking for. That being said, MasurTabachnikov's survey "Rational billiards and flat structures" and Masur's survey "Ergodic Theory of Translation surfaces" contain a lot of introductory material. Both can be found on Masur's webpage here. 


People normally take the definition: translation surface = pair (X,w), where X is Riemann Surface and w is a non identically zero holomorphic form on X. Strictly, if you want to make a translation surface out of this pair you have to remove the zeros of w from X and then integrate w. As a general reference take the book: Flat surfaces (by Anton Zorich) in collection "Frontiers in Number Theory, Physics and Geometry. Volume 1: On random matrices, zeta functions and dynamical systems'', P. Cartier; B. Julia; P. Moussa; P. Vanhove (Editors), SpringerVerlag, Berlin, 2006, 439586. 

