Is the affinity of an enzyme or transporter for its substrate or solute influenced by the amino acids at the binding site?
In a reaction which follows a saturation kinetics, KM is basically the concentration of substrate/ligand at which the rate of the reaction is half of the maximum rate (or the binding sites are half saturated).

The biophysical meaning of KM would depend on the underlying model. For example, in the equilibrium approximation of Michaelis-Menten model, KM is same as dissociation constant (of enzyme and substrate). In the Briggs-Haldane model (quasi steady state approximation), it is slightly complex: KM = (kr + kcat)/kf.

Overall, the thumb rule is that lower the KM faster will the reaction saturate.


  Is affinity always a function of the protein structure of enzyme? If
  someone study same enzyme in two different species and observe that
  the enzyme has different affinities in two species, then does it
  indicate that there might by slight difference in key amino-acid
  component at the binding sites of the enzyme between these two
  species?


Almost every property of the protein (not just affinity towards a ligand) is because of its chemical composition and structure. Although they are directly involved in binding, the amino acids at binding site are not the sole contributors to the affinity. The other amino acids that are critical for the overall structure of the protein are also important in preserving the function.