What does it says the transconductance efficiency (gm/Id) in an amplifier?
Transconductance efficiency basically measures how much transconductance is produced for a given unit of bias current (as @mkeith correctly suggested).

You can also define it for a BJT common emitter as \$g_m/I_c=1/U_T\$ (so it's the inverse of thermal voltage), i.e. around 38.6 S/A at room temperature. Typically for a properly designed BJT it is constant (and high) over several decades of collector current.

MOSFETs aren't so nice. The transconductance efficiency of a MOSFET is alway less than that of a BJT and drops significantly in the strong inversion region. Giving a summary of all that happens is a bit more than I signed up for here, but the essence is:


  The presence of effectively five regions or subregions of drain current
  operation, which also controls transconductance, illustrates the complexity of analog CMOS design compared to bipolar transistor design, where a single region of exponential operation exists over as many as 10 decades of collector current.


As a numerical example, for bulk CMOS processes at room temperature, transconductance efficiency drops from approximately 25.7 S/A in weak inversion to 8.9 S/A at the onset of strong inversion, which is 35% of the weak inversion value, and it continues to drop after that.

You'll really want to read Binkley's 2nd chapter for all the details, equations for every MOSFET region etc.