What happens if gravity of all objects in the universe disappears? I've been trying to find the answer to this question for a few days and I ended up in having different answers from different sources. A website says:

Everything on the universe would move in this exact direction it was moving at the instant gravity stopped.

And another one says:

Since it is gravity that binds planets in shape, when gravity disappears, planets will lose their shape are turn into dust clouds.

Now which of the above is correct? 
 A: The Earth is rotating at around 1000mph. There is also immense pressure inside the earth pushing the crust outwards. If gravity "stopped working" then the rotational velocity plus the internal pressure would rip the planet apart pretty quickly. However, if gravity "stopped" the sun would explode, as would all other stars, so it's debatable which would get us first - the earth falling apart or the sun burning us to a crisp.
BUT, gravity is a result of mass. If gravity "stopped working" then presumably that would be a result of all mass suddenly disappearing. The only way that could happen is if all the mass in the universe suddenly converted to energy, so the result is again a huge explosion.  :-) 
A: 

Everything on the universe would move in this exact direction it was moving at the instant gravity stopped.

And another one says:

Since it is gravity that binds planets in shape, when gravity disappears, planets will lose their shape are turn into dust clouds.


Both are correct. One is a subset of the other.
Orbiting bodies (pretty much everything in the universe orbits something else) will stop orbiting and continue in straight lines. So will things not firmly attached to whatever rock they are on. So in our case everything at the equator will leave in a straight tine tangent to their previous position, with the globe at the south pole one of the last to go. It might even stay there, depending on the adhesive action of the ice after the atmosphere departs.
Rocky planets (like ours) will certainly say goodbye to their atmosphere, surface water, and anything not firmly attached. After some most impressive volcanism I suspect there will be a rather sizable chunk of nickel-iron still holding together and slowly cooling.
Gas bodies (including the sun) will simply dissipate into a disc. The sun itself (and all other stars) will rapidly extinguish themselves - fusion requires a lot of heat, stars maintain the heat through pressure, and if gravity goes poof nothing will be maintaining the pressure anymore.
One benefit of losing gravity will be we can end all those arguments about what's inside a black hole - it won't be black anymore. Too bad nothing will live long enough to care.
A: Everything on the universe would move in this exact direction it was moving at the instant gravity stopped.
Imagine a few galaxies, separated by say 20 000 light years from each other.  When gravity is present, they pull inwards towards each other. If gravity was turned off, they would still continue on the path they were on, because there is nothing to slow them down.
But, if there is no gravity, the galaxies themselves would start to fall apart, with each star continuing in a straight line, rather than it's previous curved course around the galaxy centre of mass. 
Since it is gravity that binds planets in shape, when gravity disappears, planets will lose their shape are turn into dust clouds.
As far as planets go, there are other forces, such as the electromagnetic force, which is far stronger than gravity at close range, so atoms and molecules (and us) would still stay together as units, although moving away from Earth , as John refers to in his comments above.
For the Earth, we would lose the moon, it would head off at a tangent, but planets would not "fall to dust". 
EDIT Roland's answer  regarding the Earth, is  correct (and mine is wrong) but my guess is we would go back to the state the primordial solar system was in, (leaving the sun out of it , just for the moment, as we would be doomed anyway), with large chunks of rocks moving around in totally chaotic orbits. Although the Earth would fall apart,  it would not be to the level of dust. END EDIT
A: Gravity is an effect. There is theoretical and an interpreted experimental results that point to the concept that gravity is a result of neutrino refraction through matter. if this is true then the universe is comprised of a mat of neutrinos in constant intertwining motion. Which means IMHO that the rug got pulled out from under us and we would all disappear if gravity were to 'disappear'. 
a good analogy of gravity is to have two hoses or sprays facing each other with a porous material like a window screen between them. the water flow are the neutrinos, the screen matter. if the flows are carefully regulated then the screen will not move. this is analogous to 'zero gravity'. now place another screen between the existing screen and a nozzle, the screens are forced together. and that is gravity.   
