Short answers, pulled from the story.
Orbital resonance occurs when orbiting bodies exert regular, periodic gravitational influence on each other because their orbital periods are related by a ratio of small whole numbers. The effect is similar to pushing a swing at its natural frequency: each timed gravitational kick builds on the last, cumulatively altering or constraining the bodies' orbits.
The Laplace resonance is a three-body mean-motion resonance with a 1:2:4 orbital period ratio. Jupiter's moons Io, Europa, and Ganymede are the classic example: for every orbit Ganymede completes, Europa completes two and Io completes four. The resonance was discovered by Pierre-Simon Laplace and makes a triple conjunction of all three moons geometrically impossible.
Pluto and Neptune are in a 2:3 mean-motion resonance, meaning Pluto completes two orbits for every three Neptune completes. This timing ensures Neptune is consistently far away, averaging a quarter of its orbit distant, whenever Pluto approaches its closest point to the Sun. The minimum separation between the two bodies over two Pluto periods is 17 AU.
The Kirkwood gaps are cleared by mean-motion resonances with Jupiter at the 4:1, 3:1, 5:2, 7:3, and 2:1 positions. Asteroids that drift into these resonance zones have their orbital eccentricities steadily increased by repeated gravitational perturbations until they are ejected from the belt entirely.
Saturn's axial tilt of 26.7 degrees is thought to have been amplified by a secular resonance between Saturn's rotational axis and Neptune's orbital axis, both of which have precession periods of about 1.87 million years. As the Kuiper belt depleted and Neptune's precession rate slowed to match Saturn's, the resonance captured Saturn's spin axis and increased its obliquity from a tilt closer to Jupiter's 3.1 degrees.
Several extrasolar systems contain resonance chains. Kepler-223 hosts four planets in a confirmed 8:6:4:3 orbit ratio, the first confirmed four-body resonance. TRAPPIST-1 has seven near-Earth-sized planets forming the longest known near-resonance chain. K2-138 contains five planets in an unbroken near-3:2 resonance chain with periods of 2.353, 3.560, 5.405, 8.261, and 12.758 days.