Short answers, pulled from the story.
Planetary migration is the alteration of a planet's orbital parameters, especially its semi-major axis, through gravitational interactions with a disk of gas or planetesimals surrounding its star. The process can move planets inward toward their star or outward to more distant orbits. Migration timescales range from millions of years for gas-disk-driven migration to billions of years for tidal migration.
Hot Jupiters are giant exoplanets with Jovian masses but orbital periods of only a few days, placing them so close to their stars that standard planet formation theory says they cannot have formed there. Type II disk migration provides the leading explanation: a Saturn-mass or larger planet opens a gap in the surrounding gas disk and spirals inward as the disk evolves, potentially ending up in a very close orbit.
Type I migration applies to smaller planets that do not open a gap in the gas disk; it is driven by torques at Lindblad and co-rotation resonances and tends to be faster. Type II migration applies to planets massive enough to open an annular gap, roughly from the mass of Saturn upward, and the planet's orbital evolution then follows the viscous evolution of the disk, typically at a slower rate. The transition between the two regimes is generally smooth.
Outward migration of Neptune is believed to be responsible for the resonant capture of Pluto and other similar bodies, called Plutinos, into the 3:2 mean-motion resonance with Neptune, meaning they orbit the Sun twice for every three Neptune orbits. This happened through planetesimal-driven migration, in which gravitational encounters with a large number of small icy bodies gradually pushed Neptune outward.
After roughly 500-600 million years following the Solar System's formation, about four billion years ago, Jupiter and Saturn crossed the 2:1 orbital resonance, in which Saturn completed one orbit for every two of Jupiter's. That resonance crossing increased the eccentricities of both planets, destabilized the orbits of Uranus and Neptune, and triggered close encounters that sent Neptune plunging into the dense planetesimal belt, scattering most of those bodies inward.
Kozai cycles are a gravitational mechanism in which a distant companion star causes a planet's orbital eccentricity and inclination to exchange repeatedly, cycling through high and low values over time. If eccentricity climbs high enough, the planet's closest approach to its star becomes small enough for strong tidal forces to act, shrinking the orbit. This combination of Kozai cycles and tidal friction can also flip a planet's orbit into retrograde.