What is an Intuitive Explanation of the Phase of a Signal
Compare $y= sin(\omega t)$ to $y=sin(\omega t-\phi)$ as shown in the plot below. $\phi$ is the additional phase term in radians where $\omega$ represents the frequency in radians per second. Thus the phase term shifts a sinusoid along the horizontal axis. So at a given frequency this will result in a time delay for that frequency, although to be noted that a fixed time delay will cause a phase shift that is proportional to frequency as a linear variation (The phase will increase linearly as the frequency increases). Thus our interest in "linear-phase" filters as this will have a constant "group delay" meaning the group of all frequencies in a waveform will be delayed the same amount in time, and thus not cause destructive interference between the different frequencies in the group. (Review Fourier Series and decomposition of waveforms into individual frequencies and the resulting reconstruction for more insight on that).



More pertinent (and useful) is complex signal representation with $Ae^{j \phi}$ representing the magnitude A and phase $\phi$ of a signal. This can be expanded using Euler's identity which may help give further insight: $$Ae^{j\phi}=Acos(\phi)+jAsin(\phi) = I + jQ$$

From which we see that in order to completely describe a complex signal we require two real signals, this could be magnitude and phase, A, $\phi$ as in $Ae^{j\phi}$ or the real and imagninary components I, Q as in I+jQ. 

Refer to this post which may help give further insight into the importance and utility of complex representation for signals: Frequency shifting of a quadrature mixed signal