Time domain,Freq. domain,phasor domain
The time domain is pretty easy. \$ I_0  sin(wt) \$ The signal varies with time \$t\$ with the frequency \$w\$. The problem with this notation is that i's difficult to deal with delays and comparing the original signal to the delayed one \$ I_0  sin(wt - T) \$ for example. The solution is to transform the signal to the phasor domain. This gives the ability to compare signal timing in degrees instead of time. For example two sine waves that are 360 degrees out of phase line up again. The downside of straight phasors is that every signal in the analysis must be at the same frequency and there is no compensation for components that are frequency sensitive (inductors and capacitors). 

The frequency domain transforms all frequency sensitive components to compensate them for frequency. The frequency domain still uses phasor notation. If frequency domain you can also deal with multiple frequency signal but to do that you must use super position. 

To move from time to frequency just use a Laplace transform. To move from time to phasor figure out how much delay or advance the signal has \$ degress/360 = T/period \$