Halo orbit families exist near the L1, L2, and L3 librations points. This video focuses on the L1 and L2 halo families. There are northern and southern families at each of the libration points. The northern family is identical to the southern family but mirrored across the x-y plane.
At each point, the family bifurcates from the planar Lyapunov family of orbits. That is, the first halo in the family is planar, and it is also a member of the kidney-bean-shaped Lyapunov family. You can step up in the z direction and find another member of the halo family. And step up again to find the next. The family thus evolves out of plane, as you see in the video, and it keeps growing and approaches the smaller primary (the Moon if we're working in the Earth-Moon system). The Near Rectilinear Halo Orbits are the tall, nearly-polar ones that get really close to the Moon. How are they defined?
We've defined the NRHOs as those members of the halo families whose stability indices are bounded. That is, they're marginally stable, or nearly so, in a linear analysis. You can see a plot of the stability indices of the L1 and L2 halo families, a zoom of the NRHO portions of the two families, and the butterflies as a bonus, in Figure 2 of this paper:
https://engineering.purdue.edu/people/kathleen.howell.1/Publications/Conferences/2017_AAS_DavPhiHow.pdf
Check out the top plot in Figure 2b. See how the stability indices are bounded (with a value of 3 or less) until you hit a perilune radius (rp) of 16,000 or 18,000 km, and then the stability indices begin to grow quickly? We've defined "NRHO" to lie to the left of that bifurcation- those halos with bounded stability properties. There's another bifurcation on the far left of those plots, where the stability index is equal to 1. You can see it for the red L2 line; for the blue L1 line that point is below the lunar surface and isn't included in the plot. That bifurcation marks the lower limit of the NRHO portion of the halo families.
Why are they special? As discussed in Ryan Whitley's paper you linked to, they're favorable for transferring into and out of, which is nice when your spacecraft is meant to be a staging ground for exploration, with other ships coming and going. They also provide really great lunar south pole coverage, since they whip quickly around the north pole and spend almost all of their time in the southern hemisphere. See: https://engineering.purdue.edu/people/kathleen.howell.1/Publications/Journals/2008_JSR_GreOziHowFol.pdf
In addition, they're close to stable. In a flatter, less stable halo, if you miss a stationkeeping maneuver or have an unexpected perturbation, you might escape from the halo in just a few weeks or less. In an NRHO, the lower stability index means that a missed maneuver or a perturbation affects your orbit less, and there's more time to recover before the spacecraft escapes the orbit. So- orbit maintenance (or stationkeeping) is required, but there's more recovery time than in a flatter Earth-Moon halo like Artemis flew.
Illustration for clarity:
Figure 2b from this paper:
