One of the known method for this is Reduced gravity aircraft , unofficial nickname vomit comet .
KC-135A known as NASA 930 was used in the movie Apollo 13 .
Cecil said that in 2001: A Space Odyssey , huge rotating sets are used, while the actor remained more or less stationary.
It's not that realistic in looks but cheaper as compare to vomit comet.
Wire and green screen combination are also used sometimes.
movies.stackexchange.com/users/1190/Ankit Sharma
Inception is well known for simulating all of its zero/reduced gravity scenes without CG.
There is a behind the scenes documentary where the effects team detail the various method they use – and crucially how different shots within a sequence use different methods.
This means there's never a single "trick" that the eye can catch to break the immersion.
Christopher Nolan:
We did it through a number of different rigs and in the final edit what you see is shot to shot to shot it tends to be a different orientation and a completely different rig in each shot
and I think that more than anything else really stops the audience of seeing the trick of how this scene is done.
They used different sets for the scene, with different orientations, suspending the actors on wires to let them "fly" through the scene.
The long hotel corridor was built upright, so that the walkway was vertical and the camera is looking upward from the bottom.
Then they hung the actors upside down so they can freely move towards the sides of the corridor.
Paul Franklin:
The vertical corridor is supposed to be the same location as the horizontal corridor, it's an identical set.
The difference is that it has been built vertical standing on its end.
This means we drop actors and stunt performers on wires down into the set and the camera looks up at them.
They can then be raised and lowered and swing around the sides and it looks like they're floating in zero gravity.
The actors also tried to simulate the behaviour of weightless people, based on real examples.
Joseph Gordon-Levitt:
In real zero-G, and I've actually spoken to people that have been in real zero-G, what they told me is they never felt so relaxed in life.
What I did is the exact opposite of that.
In order to make it look like that was the case I actually had to keep every muscle tight because I was supporting myself.
movies.stackexchange.com/users/6496/Gareth
The recent film Gravity used motion control rigs adapted from automotive assembly line robots.
Popular Mechanics did a pretty interesting article about it: http://www.popularmechanics.com/technology/digital/fact-vs-fiction/free-floating-feel-how-gravity-simulates-zero-g-16016504
movies.stackexchange.com/users/6501/Cody
More recent films also use digital doubles to animate all but actors' faces.
This was used extensively on Ender's Game, and to some extent on Gravity.
Per
this article : To create a sense of weightlessness, the filmmakers shot actors in harnesses and on wires against a greenscreen set to place them in a virtual world.
Then to achieve the look of zero-G — and obey the laws of physics — visual effects studio Digital Domain ultimately retained only the actor’s faces and relied instead on digital doubles of their bodies for the sequence.
And for much more detail : Nearly all the digital doubles created for Ender’s Game were in the zero-G Battle Room. . . .
Digital Domain’s animation team developed tools that allowed artists to correct for that movement, then re-projected digital doubles (or parts of their bodies) back into the shots.
Because Digital Domain had developed CG versions of the actors’ flash suits and re-created the lighting environment digitally, they were able to keep the actors’ faces from the live action shoot and
replace nearly all of the body motions with digital doubles.
movies.stackexchange.com/users/5352/Geoffrey Booth
If you are looking for a low-budget option, it is worth doing a YouTube search for Anne Hathaway's zero-gravity move on the Graham Norton Show.
She demonstrates how, when filming Interstellar, actors were required to simulate weightlessness by standing on one leg.
She stands on one leg with a relaxed knee and extends the other leg behind her so that it looks like it is floating on camera.
It is a surprisingly effective technique, with zero-budget.
movies.stackexchange.com/users/29228/babkb
One answer "Apollo 13
" Space Scenes Weren't Hollywood Magic.
And There's No Such Thing As A Weightless Chamber.
So How Did Tom Hanks, Bill Paxton And Kevin Bacon Float?
They Filmed
Aboard NASA's "vomit Comet."
movies.stackexchange.com/users/18249/Gerald Edgar
Director Ron Howard describes other techniques in his commentary for Apollo 13 for simulating weightlessness in scenes shot on the soundstage.
The actors often sat one one end of a see-saw which was weighted on the other end, allowing them to bob up and down smoothly.
Simple camera tricks were used such as having the actors lay down on their sides opposite each other, and turning the camera 90 degrees to making it appear that one actor was completely upside down.
Also, they filmed the scenes in the KC-135 before the soundstage ones so the actors knew what real weightlessness looked and felt like, and they could more easily fake the look on earth.
movies.stackexchange.com/users/30850/Drew C
How can the weightlessness of space be  realistically  simulated on film? Does it require CG or can a convincing simulation be achieved using  wire fu  and editing?  (I'm happy to hear of good examples of weightlessness in movies and how the effects were achieved.)
How can weightlessness be simulated on film?
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movies.stackexchange.com/users/3650/coleopterist
Director Ron Howard describes other techniques for simulating weightles sness in his commentary for Apollo 13. Actors often sat on one end of a see-saw which was weighted on the other end, Simple camera tricks were used and they also filmed scenes in the KC-135 before the sounstage so the actors knew what real weightlessness looked like.  The actors often sat one one end of a see-saw which was weighted on the other end, allowing them to bob up and down smoothly. Simple camera tricks were used such as turning the camera 90 degrees to make it appear that one actor was completely upside down.
 Director Ron Howard describes other techniques in his commentary for Apollo 13 for simulating weightlessness in scenes shot on the soundstage. The actors often sat one one end of a see-saw which was weighted on the other end, allowing them to bob up and down smoothly. Simple camera tricks were used such as turning the camera 90 degrees to make it appear that one actor was completely upside down. They filmed the scenes in the KC-135 before the soundstage ones so they could more easily fake the look on earth.