What does "electrochemically active species" mean in the expression for the mass-transfer limited current? The expression for the mass-transfer limited current $i$ is given by
$i = nFAmC$. 

where $C~[\mathrm{mol\, cm^{-3}}$] represents the concentration of the electrochemically active species that gets converted at the electrode; if the entire analyte gets converted, then $C$ is equal to the initial bulk concentration of the analyte.

I am confused as to what electrochemically active species mean? Is that just saying the ions in solution? Or is it the oxidized ion and the reduced ion?
It is the species that can get oxidized or reduced in the system. 

It is specifically written out in order to make sure that it doesn't get confused with supporting electrolyte or any other component in the system.

So, for a reduction reaction, it is the concentration of the oxidized species.