Mid-Air Bobbing

Objects and people that hover in the air will always, always bob up and down, as if floating on some invisible surface of water. This is typically because being perfectly motionless gives the unpleasant impression of being "stuck" in the air. The floating is often accompanied with a whoosh sound, for no other reason than the Rule of Cool.

The most egregious instances of this trope are when objects in zero gravity behave like this - bobbing up and down despite the fact that 'down' doesn't exist here. This conclusively proves that Space Is an Ocean.

Selective Gravity causes many items in video games to do that.

Justification usually invokes one of the following:

It is possible that the bobbing is merely the character performing drift correction. Satellites in orbit, for a real life example, have thrusters to keep them from drifting out of the required orbit all the time.
In real life, of course, few things float in midair, and those that do (such as tethered or semi-deflated helium balloons) tend to bob in all but the stillest air—or to be surrounded (as a hummingbird or helicopter) by a blur of corrective motion. And, of course, things floating on water bob up and down too. So this trope seems to derive from carrying learned expectations about the natural behaviour of objects over into extrapolations about unnatural behaviour.
In animated cartoons, especially lower budget ones, midair bobbing serves the pragmatic purpose of establishing that a character or object is supposed to be in the air, and isn't just misaligned with the background.

Examples of Mid-Air Bobbing include:

Anime

Urd in Ah! My Goddess likes to do this when "sitting" rather than sitting on furniture. She takes after Hild, who probably expends less energy at it than you would on breathing, being the equal and opposite number of that verse's God.

Film

Luke Skywalker and the other X-Wing pilots do a good bit of this in Star Wars. You can see it, for example, in this Youtube clip. It's only when they do a close-up cockpit shot of one of the X-Wing pilots. Everybody just kind of...bobs. Very odd, especially since the X-Wings themselves certainly aren't bobbing around in the longer shots. The speeders largely subvert this however in that when they are idling they are completely motionless and often don't even react when someone is mounting/dismounting/operating them (aside from Luke's speeder in ANH, but that is mentioned on screen to be a bit of a cheapo/junker.)
In Apollo 13, the bobbing was due to the set (which was in the Vomit Comet) being buffeted by the atmosphere.

Literature

Peeves in Harry Potter. Most likely not due to the qualities of floating, but due to the fact that he has the energy and attention span of a ritalin deprived ADHD 6 year old on a sugar rush.
- Many other things float without this quality. Snape (when being levitated by Sirius) does bob, but that's more due to Sirius's lack of fine control.

Video Games

Any powerup, item, or other object that floats in a videogame (especially a newer one) will bob up and down, often also slowly spinning as if on an invisible store window display. The main reasons are "highlighting" of interactive objects (because Pixel Hunt was only occasionally amusing back in the era of 640x480 and no textures), and the fact that in a non-stereoscopic image a small object without texture that clashes with everything else too often isn't recognizable or even noticeable from wrong angles.
Anyone affected by the "Float" spell in any given Final Fantasy game will hover above the ground for a while. Flying enemies also did this in IV through VI.
In Pokémon: Mystery Dungeon games Lt. Magnezone and several other fliers and levitators do it.
The player in Descent. Since you're in a spaceship, it's presumably drift correction.
The Allied Rocketeers in Command & Conquer: Red Alert 2 do this.
Happens in City of Heroes when the hover power is engaged and the hero isn't otherwise moving. The official website header makes it even more obvious.
RPG Maker: Giving an enemy the 'flying' attribute will cause it to midair bob in battle.
Halo: The thrones of the Prophets and Rtas 'Vadum (aka Half-Jaw) both behave this way.
Watch the ships in Wipeout HD when at a standstill. As one would reasonably expect from machines designed to perform best at very high speeds, the ships appear to be performing drift correction to avoid shifting sideways as well as bobbing naturally with air currents, especially close to the ground, and when they move faster, they stop doing this-presumably due to their shapes enacting aerodynamic stability on the ships, allowing them to fly without wiggling everywhere.
The fire pets from the "Firemaker's Curse" quest in RuneScape do this.
Late Wizardry games have it as floating and flying creatures like all other go through idle animations. It also serves to convey information, as the animation freezes when a creature is paralysed or knocked out.

Web Original

In one Strong Bad email, Bubs is shown to have once possessed the power of flight. He bobbed.

Western Animation

The Powerpuff Girls are the queens of this trope: not only do they perform Midair Bobbing, they do it in unison.
Fairy Godparents in The Fairly OddParents all bob like this. And they all do insync, as if floating on the same invisible plain.
- And so do some of their buildings.
As do both Starfire and Raven from Teen Titans. But mostly Starfire.
The hoverbikes in The Venture Brothers.
Happens in the episode of Chip 'n Dale Rescue Rangers where they launch into space on a spaceship. This one is particularly silly; while whatever Applied Phlebotinum is used in the other examples may produce Mid-Air Bobbing, physical weightlessness definitely doesn't.
All over the place in The DCAU. Occasionally averted, which tends to look weird.
Happens in Avengers Earths Mightiest Heroes when Iron Man flies into space to search for Big Bad Kang's ship. Of course, this was right after coming to a stop without benefit of retro-rockets.

Real Life

There is a similar phenomenon in aerospace called a "pogo oscillation", a reference to the bouncing motion of a pogo stick. It results when a problem with the fuel system causes fluctuations in the fuel flow rate, causing likewise fluctuations in engine thrust output, causing likewise fluctuations in acceleration. This fluctuation in acceleration worsens the fluctuations in the fuel flow rate, forming a vicious circle that eventually leads to a structural failure in the vehicle and, in some cases, complete and total vehicle disintegration.