Homology and years after Cohomology became the main object of study for many areas of mathematics the last 150 years or so. The reason is that they both provide a very powerful machinery for classifying objects (especially geometrical like topological, differentiable manifolds, CW complexes, Schemes, Varieties etc.) but each one from a different point of view. Whilst the above terms are referring usually to the so-called singular (co)homology theory these two notions have been expanded through Category Theory as a tool in many other areas like Algebraic Geometry, Algebraic Topology, Simplicial Homotopy Theory, Homological Algebra etc. In this talk we will focus mainly on understanding the basic idea behind those things through some explicit calculations and the uniqueness of such a theory via the Eilenberg-Steenrod axioms.