A vector is a line segment that has both length and direction (i.e., the direction is indicated by an arrow head). You can use a vector to model anything that has both of those characteristics. For example, you can use a vector to model the distance and direction you would have to walk, to get from your house to the bus stop.
A phasor is a vector that has been used to model a physical parameter that varies like a sine wave. All phasors are vectors, just like all cats are animals, and the reverse is not true in either case.
For example, we use phasors to model currents and voltages. The length of the phasor can represent the peak value of a voltage, though we usually have it represent the RMS value. The direction (i.e., angle with respect to the horizontal) can represent the phase angle of one voltage, as compared to another. For example, in a balanced 3-phase system, I would draw three rays of equal length, with 120 degrees between any two. I would lay the first one out horizontally on the paper, pointing to the right, and call it "Voltage of Phase A." The ray pointing down and to the left would be the "Voltage of Phase B," and the one pointing up and to the left would be "Voltage of Phase C." The sequence can be visualized by imagining the three rays spinning in a circle around the center point. The spin is counter-clockwise. So if you stand off to the right of the diagram, you will see the three rays pass by you in the sequence A, then B, then C, then A, and so forth.
You can model an unbalanced 3-phase system by having different lengths for the three phasors, and by having the angles between them be something other than 120 degrees. However, you cannot use phasor diagrams to model something that is not a sine wave, or for which all parameters do not have the same, constant frequency. You can't model harmonics with phasors, except by having each harmonic (i.e., the fundamental frequency, the second harmonic, the third harmonic, etc.) in its own diagram.