Why objects are uniquely defined by their right ascention and declination? I've just started studiyng Astronomy.

I understood how I would measure Right Ascention and Declination: for the first one I'd trace an Hour cirlce to the Celestial Equator, and then, compute the angle from this point and the Vernal Equinox; while the second one is the angle subtended by the object and the intersection of the Hour Circle with the Celestial Equator.

So, during the night, stars will change their Right Ascension value? Does this means this coordinate system is not fixed with respect to the stars?

If so, why people store the RA/Dec to find objects?
For instance, I know  the RA/Dec of the NGC 5585 Galaxy ( http://en.wikipedia.org/wiki/NGC_5585 ), but how can I use this information, if the RA changes during the night?
Stars do not change their right ascension or declination values as the 
Earth turns. 

The position of the stars in the night sky (called azimuth and 
elevation) will change, but that's because the lines of right 
ascension also move. 

However, right ascension and declination do not uniquely define an 
object. Over time, stars move with respect to the sun (called "proper 
motion"), and the direction of the Earth's north pole (currently 
pointing near Polaris, the North Star) also changes (called 
"precession").