What distinguishes quasi-satellites from true satellites?
A quasi-satellite is an object in a specific type of co-orbital configuration (1:1 orbital resonance) with a planet where the object stays close to that planet over many orbital periods.
A quasi-satellite's orbit around the Sun takes exactly the same time as the planet's but has a different eccentricity (usually greater). When viewed from the perspective of the planet, the quasi-satellite will appear to travel in an oblong retrograde loop around the planet.
In contrast to true satellites, quasi-satellite orbits lie outside the planet's Hill sphere and are unstable. Over time they tend to evolve to other types of resonant motion, where they no longer remain in the planet's neighbourhood, then possibly later move back to a quasi-satellite orbit, etc.