Does lowering arduino analog reference increase precision in measuring low voltage? The definition for Analog Reference is as follows: The reference voltage used for analog input (i.e. the value used as the top of the input range).

By default you have analog reference of 5V. So the precision of analogRead is 5 / 1023 = 0.0048875.

Now what happens if I lower the analog reference to let's say 1V. Does that mean that the precision now is 1 / 1023 (and the arduino can't measure voltages higher than 1V) or it is the same precision as before with the only difference being the fact that the arduino can't measure voltages higher than 1V anymore?
The former, i.e precision would now be 1/1023 but the accuracy of the AD converter is 4 / 1024 or 1 part in 256  i.e. 0.25%. 

Also be aware that the internal reference voltages themselves are not very accurate : 

DEFAULT: depends on your computer power supply USB2  Vbus = 4.4V - 5.25V, Vint 1 = 1.1V  actually 1.0 -- 1.2V, Vint 2 = 2.56V  actually 2.4 -- 2.8V