NEVER apply 12V to an LED in the REVERSE polarity! That is way beyond specs and is practically guaranteed to destroy the device! Also, NEVER apply any voltage directly to the LED in either polarity without a current limiting resistor of some sort, even a voltage as low as 2 Volts with "unlimited current" will destroy the device! Finally, most LEDs are intended to be operated from a DC supply and and will not survive AC.
That being said, generally 3 to 5 volts in the reverse direction is considered safe (two 1.5V batteries in series or one of the
3V coin cells will do the trick. A 1K (1,000 ohm) resistor will limit the current to just a couple of milliamps and will be enough to light the LED in the forward direction. A 100 ohm resistor from 5V will limit current to around 30 milliamps and is generally beyond the specs for most LEDs.
If the LED is specified to have an internal current limiting resistor and you have determined the proper polarity you can
apply various voltages. The brightness is directly proportional to the current. For any given current limiting resistor you
can vary the applied voltage to vary the brightness. Likewise, for any given supply voltage you can vary the value of the current limiting resistor to vary the brightness.
For LEDs without an internal current limiting resistor there is a "Forward Voltage" specification typically ranging from around 1.5 to 3.0 volts. (Red is generally in the area of 1.5 volts). If you're running from 12VDC you can calculate the resistor value by subtracting the FWD voltage spec from the 12V and divide by the desired current. Conversely the applied voltage minus the FWD voltage spec and divided by the resistor value will give you the resulting current.
Generally speaking, a doubling of current will double the brightness, about one f-stop. From a 12 volt supply one LED will cause a certain brightness for any given current limiting resistor. Two LEDs in series will see only a small variation in current with that same resistor value. (12V-1.5V)/resistance -vs- (12V-3V)/resistance is only a 10.5:9 ratio. Even 4 LEDs in series would only result in a 10.5:6 ratio. From a 5V source, two LEDs in series would vary the current for any given resistor value by 3.5:2 and you would need well over 6V to light a 4 LED series string.
Resistor values of 220, 470, 1K, 2K, and 4.7K are common and will give you a wide range of brightness from any LED and a variety of supply voltages. Be careful with the lowest two values, they can cause excessive current with just one LED from a 12V supply.