Short answers, pulled from the story.
Orbital resonance occurs when orbiting bodies exert regular periodic gravitational influence on each other so their orbital periods relate by ratios of small integers. This physical principle works like pushing a child on a swing where the body doing the push acts in periodic repetition to have a cumulative effect on the motion.
Pierre-Simon Laplace found the first answers explaining the linked orbits of the Galilean moons after Newton published his law of universal gravitation in the 17th century. Before Newton there was consideration of ratios and proportions in orbital motions called musica universalis or the music of the spheres but Laplace provided the explanation for these stable orbits.
The dwarf planet Pluto follows an orbit trapped in a web of resonances with Neptune including a mean-motion resonance of 2:3 that ensures when Pluto approaches perihelion Neptune is consistently distant averaging a quarter of its orbit away. Other much more numerous Neptune-crossing bodies that were not in resonance were ejected from that region by strong perturbations due to Neptune.
In the rings of Saturn the Cassini Division is a gap between the inner B Ring and the outer A Ring that has been cleared by a 2:1 resonance with the moon Mimas. The Encke and Keeler gaps within the A Ring are cleared by 1:1 resonances with the embedded moonlets Pan and Daphnis respectively while the A Ring's outer edge is maintained by a destabilizing 7:6 resonance with the moon Janus.
TRAPPIST-1 has seven approximately Earth-sized planets in a chain of near resonances forming the longest such chain known with nearest-neighbor period ratios proceeding outward of about 8/5, 5/3, 3/2, 3/2, 4/3 and 3/2. Each triple of adjacent planets is in a Laplace configuration such as b c and d in one such Laplace configuration and c d and e in another.