analogReadResolution(bits) sets the bit width of the values which you get with analogRead(pin).
Thus, if bits = 10 (default), the values you can obtain with analogRead() will go from 0 to 1023.
If you change the resolution to 12, you will obtain 0 - 4095, with 6 you will obtain 0 - 63 and so on.
I general, setting the resolution to n bits, n in [1..16], you will read values from 0 to (2^n) - 1.