What is the absolute phase and the relative phase, of a signal?
"Absolute phase" is really relative phase to a point in time at which the phase measurement is referenced (Noon GMT on Tuesday, the first sample in a time-domain window of sample data for instance, or maybe the middle sample in a vector).  
Relative phase can also be measured with respect to another waveform of the exact same constant frequency, as the difference in phase measurement between the two waveforms at any one point will be the same at all other points, as long as the two frequencies stay identical.


Phase is important because it is one of the 3 parameters needed in an equation that completely describes a sinusoid.  Without phase information, the waveform is ambiguous.

$$y(t) = a\cdot \cos\left(b\cdot t + \phi_{t_0}\right), \quad \text{ where }\phi_{t_0} \text{is the phase at time } t = t_0.$$