@Federico reminder has complicated a bit the answering process, but now we understand your actual problem, I can provide a focused answer immediately below, I've moved other elements at the end, you may ignore them as they were aimed to clarify your question and are indeed no more relevant.
If I consider Y as relative velocity, Then X should also be the same, though in opposite direction, right?
Good point, they are the same magnitude. There is some approximation in the drawing:
I am not able to understand the term defined by X
The convention used for velocities in this drawing seems to be: Velocities relative to rotors are drawn as dotted lines and velocities relative to stators are drawn as plain lines. As stators are fixed to the engine, plain line velocities may be seen as "absolute".
- Y is the tangential velocity of IGV, and stators in general, relatively to the first stage rotor, and to rotors in general. Y is the apparent velocity of the stators.
- X is the tangential velocity of R1 (first stage rotor) relatively to S1 (first stage stator). X is the true velocity of the rotors.
In the Note, the relative velocity is shown by D, which I think is similar to C, making C relative velocity
If you look at R1, there are arrows to emphasize the rotor channel is a diverging duct:
The channel slows down air and at the same time it increases pressure.
E described as "absolute" is linked to the fact it is relative to stator S1, which can be seen as being in the same reference frame than the engine, and "absolute".
- D is the velocity of the flow relatively to R1, and E is the same velocity relatively to S1. The difference is: For the stator, D is affected by the rotation X.
- For the rotor R1: C was transformed into D. D magnitude is smaller because of the deceleration by R1 diverging channel.
- For the stator S1: B was transformed into E. E magnitute is larger because of the downwash created by R1.
- As the mass is the same, and force is mass times acceleration, there is an excess of energy at the inlet of S1. This is the key to understand the compressor principle.
S1 stator channel is again a diverging duct. It converts the energy added by the rotation into pressure, slowing down air from E to F velocity:
This process ends here for the first stage, it will be repeated the same for the other stages, except IGV are not required, as the stator is designed to provide the correct angle of attack and velocity G for R2.
Bottom line, if we look at the process from the compressor standpoint, that is from the stators standpoint, for the stage R1/S1:
- Air at the outlet of the stage has the same velocity it has at its inlet, F ≈ B
- Pressure at the outlet is a bit larger than at inlet, due to addition of rotational kinetic energy (which source is the turbine, and ultimately fuel).
- Adding stages allows to incrementally increase pressure. The pressure ratio of a stage is limited by the angle of attack on blades and vanes. Increasing pressure means decreasing (axial) velocity. A smaller axial velocity at the outlet conjugated with a constant X/Y tangential velocity increases the angle of attack on the next airfoil, possibly past the stall angle.
IGV have no velocity-to-pressure conversion function, their role is only to create the correct angle of attack for R1. The equal length arrows emphasize an IGV channel is neither diverging nor converging:
If the velocity magnitude at the first stage inlet has to be adjusted relatively to the ram velocity, with such IGV, this must be done by the engine inlet duct, prior to the IGV.
The stator vanes have a double role, the one explained above for IGV, and the one to convert energy added by X into pressure.
Your initial questions are answered below.
Why the Y vector is in opposite direction?
Y velocity is the relative velocity of air for the rotor, it's opposed to the rotor actual rotation. It's like air coming from front when a wing moves forward:
How will the pressure zones (high and low) will form on the rotor and stator airfoils?
Blades and vanes are simple airfoils, which here are subject to chordwise winds C, E and G, exactly like regular wing airfoils. Pressure gradients will form like on any airfoil. E.g. for C and the first rotor blades, on the left your drawing, on the right the same drawing rotated: