An LED has a cut-off voltage of 1.6-4V. If you power a red LED (1.6 cut-off) with 1.5V, it will be completely off (well, maybe not, but veeeery dim). You need a current limiting device for an LED. The simplest is a ballast resistor. Let's say you have a 5V DC source and this red LED. You want 10mA in the LED, you put a 340 ohms res in series (5v - 1.6v = 3.4V; 10mA is 100 ohms/Volt; 3.4*100=340). You don't have 340ohms, you have 330; current wil be 3.4/340 = 10.3mA. Close enough. Now for some reason you don't have 5V, you have a fresh PP3 battery, 9.6Volts. The current will be: 9.6 - 1.6 > 8V/330ohms = 24mA. The LED will be brighter and last only 57 years instead of 114 (1 000 000 hours); big deal!
Now if you power the LED out of one AAA cell which delivers 1.6V when fresh, you may run out of luck and never exceed the cut-off voltage and just a slight temperature variation may change the battery voltage enough to enable lighting or turning it off.
In applications where a string of LED's is involved (push-button illumination or LED "bargraphs"), it is usual to implement a CCS (constant-current source) - typically one transistor and a couple of diodes, so the variation of the number of LED's does not change their luminosity.
Now there are LED drivers based on SMPS or capacitor charge-discharge, that handle the current regulation aspect.