Almost all of the non-star things we can see from Earth are gravitationally bound to our Sun. The other planets, the asteroids and comets, moons, etc are all part of our solar system, which means they are bound to the Sun. Most of the planets have nearly circular orbits, so they have very repeatable, normal patterns, while some of the comets and asteroids have highly elliptical orbits (spend a little bit of time close to the Sun moving fast, but spend most of their time far away from the Sun, moving slow). It's these comets with highly elliptical orbits that have these odd patterns you're mentioning. Probably the most famous comet, Halley's Comet has a very high eccentricity (of 0.96. 1 is the max eccentricity, Earth's is 0.016), meaning it can be up to 35 AU (1 AU is the average distance from the Sun to the Earth), and down to 0.5 AU.

What does it mean to be gravitionally bound? One way of thinking about it is that the total energy of the system (system being object orbiting and the object it's orbiting) is negative. How is it negative? Traditionally we consider gravitational potential energy to be negative- and it gets more negative the closer you get to it. Kinetic energy is positive, increasing with speed. So, if the sum of the kinetic energy + potential energy is negative, then the object is "gravitionally bound" to the object it's orbiting. This is how you calculate escape velocity, and another way of saying it is that the comets are traveling at less than the escape velocity of the Sun.