On the notion of a component in thermodynamics Wikipedia article on a component (in thermodynamics) first mentions that:


  In thermodynamics, a component is a chemically-independent constituent
  of a system. The number of components represents the minimum number of
  independent species necessary to define the composition of all phases
  of the system.


A bit later it is said in the same article that:


  The number of components is equal to the number of distinct chemical
  species (constituents), minus the number of chemical reactions between
  them, minus the number of any constraints (like charge neutrality or
  balance of molar quantities).


How does the second statement follows from the first one?
Well, a reaction comes with an equilibrium constant, i.e., mathematically speaking, some equation on constituents. Now we have N unknowns (constituents) and M equations (constraints and equilibrium constants). Supposedly N>M. This means we can't determine all unknowns. To do so, we must fix N-M free variables. These are our components.