In flight vehicle mechanics, there are 3 traditional aircraft axes for analysis and simulation: body, stability, and wind axes.
Fig. 1 - Depiction of traditional AC axes.
Let's start with the body axes. These axes represent the position the aircraft is currently in. These axes follow the typical orientation for the XYZ coordinate system of a plane where the x-axis is pointing along the fuselage and through the nose, the y-axis is pointing along the right wing, and the z-axis is pointing downward from the plane, perpendicular to the x and y axes. The body axes are used for constructing the equations of motion, the angular velocities, and the Euler attitudes.
Now let's talk about the stability axes. Notice the stability axes are simply an angle change $ \alpha $ of the pitch from the body axes. The stability axes are used for aerodynamics because the lift and drag vectors are nicely aligned with the axes which help with analysis.
Finally, we will talk about the wind axes. The wind axes are found by rotating the stability axes about the z-axis (changing the stability axes' yaw) by an angle $ \beta $. This axes represents the flow velocity vector of the wind.
Given a velocity for the plane, we can actually find the angles $ \alpha $ and $ \beta $. Remember, $ \alpha $ is the change in pitch of the plane, so we can define the angle as
[ 1 ]
$$ \alpha = \tan^{-1} \frac{v_z}{v_x} $$
Similarly, $ \beta $ is the angle of change in yaw from the stability axes, so
[ 2 ]
$$ \beta = \tan^{-1} \frac{v_y}{\sqrt{v_x^2 + v_z^2}} $$
I found the visualization in Figure 2 to be helpful for picturing this.
Fig. 2 - Visualization of traditional AC axes angles.