galaxies - Create Position-Velocity diagram from a velocity field

Such diagrams are usually called rotation curves, and show the velocity of stars/gas in a disk galaxy viewed edge-on along the line of sight (LOS) as a function of distance from the center.

That is, the $x$ axis gives the distance of the stars from the center ($x=0$) of the galaxy, and the $y$ axis gives the measured velocities at a given distance. Along a given LOS you'll find not a single velocity, but a (Gaussian-like) distribution of velocities, both because there is a certain scatter in velocities, but also because you only measure the component of the velocities projected onto your LOS, while both speed and direction change along the LOS, as seen here:

If they all had the same velocity along the LOS, the plot you show would be a thin line; instead it is smeared out.

Due to mass being concentrated in the center, there's a steep increase in velocity at low $x$. If there were only the mass you can see, the velocity would decrease at larger $x$ due to the larger distance from the central mass. The reason it stay more or less constant is thought to be the extra, invisible mass known as dark matter.

The reason the left side of the plot is similar to the right, but mirrored not only across the $y$ axis but also the $x$ axis, is that on this side, the stars move toward you, while on the right side they move away from you. The whole galaxy evidently moves away from you at $500,mathrm{km},mathrm{s}^{-1}$, since this is the velocity at $x=0$.

So, to recap: How do you generate such a plot? You measure the distribution of velocities along many lines of sight in the galactic plane, from one side ($-R_mathrm{gal}$) to the other ($+R_mathrm{gal}$). For instance, at $x=14,mathrm{kpc}$, you'd measure $V=675pm12,mathrm{km},mathrm{s}^{-1}$:

In practice, images like this is made by using a grism which disperses the light according to its wavelength, together with a slit placed along the galactic plane, such that you block the light from the rest of the field. This is called slit spectroscopy, in contrast to "normal" spectroscopy where you block all light except for a little hole placed on some object (e.g. a star), or integral field spectroscopy, where you get the whole spectrum behind every pixel in an image.