"Natural units" of mass Gravitational attraction is given by $\frac{GMm}{r^2}$ while attraction due to electric charge is given by $\frac{q_1 q_2}{r^2}$. Why does gravity need a constant while electric charge doesn't? Because we've picked "the right" units for charge so that the constant is 1.
This is my vague understanding of "natural units". My question is: what would natural units for mass look like, and why don't we use them?
 A: Well, that equation for the force due to electric charges is only true for a very special choice for the unit of the electric charge.  Typically, you would write down Coulomb's law as
$k\frac{q_{1}q_{2}}{r^{2}}$, where $k$ is a constant of proportionality chosen to make the units work out.  IN the SI system, the unit of charge is the Coulomb (C) and the value of $k$ is approximately $8.99 \cdot 10^{9} \frac{N\cdot m^{2}}{C^{2}}$.  
If you choose to express your charge in Gauss units, defining one Gauss to be numerically equal to $\sqrt{8.99\cdot 10^{9}} \,\,C$, then the value of the Coulomb constant becomes 1 in this new system of units.
Now, you could do the same thing for gravity, but your unit of mass would be really small: $\sqrt{6.67 \cdot 10^{-11}} \,\, kg$.  It would also mess up the definitions for any unit that had dimensionality of mass, like the Newton, Joule, etc.  But you could do that.  
Also, there is a scheme called Planck Units that attempts to set as many fundamental constants of nature as possible equal to one.    
