Letting user choose from 2 options for QGIS Processing script input parameter?

If you are not limited to QGIS v2.8 (your screenshot points to v2.8 documentation), you can install a newer version. For example, in QGIS v2.14 you have a selection parameter type available (see the docs):

You can use it in this way in the header of the script, separating options with a semicolon:

##Units=selection Millimeter;Centimeter;Meter

And later in the script you can get the chosen value in this way:

if Units == 0:
    # User chose millimeter
elif Units == 1:
    # User chose centimeter
elif Units == 2:
    # User chose meter

For reference, here you have an example script.

Germán Carrillo's answer is the most convenient. A possible alternative (if you're stuck with QGIS 2.8) could be to:

• Use a string which allows the user to enter the units they want to use:

  ##units=string mm
  
  if units == 'mm':
      # Do something
  elif units == 'cm':
      # Do something
  elif units == 'm':
      # Do something
  

  

---

• Or use three boolean checkboxes:

  ##Millimeter=boolean
  ##Centimeter=boolean
  ##Meter=boolean
  
  if Millimeter == True:
      # Do something
  elif Centimeter == True:
      # Do something
  elif Meter == True:
      # Do something