They would float and they would stop moving.
If gravity were to cease, they would cease orbiting their sun. Their inertia would continue, sending them on a straight-line path in the direction they were traveling at the moment gravity stopped. If the planets themselves had no gravity, they would eventually start breaking apart, because gravity is probably the biggest single force holding large masses together. Magnetic forces would keep certain solid parts clumped together, but a planet with an atmosphere, for example, would lose the atmosphere completely.
If there was no gravity, the Sun and and the planets would never have formed.
Planets can't really get out of their orbits because of gravity; if gravity somehow stopped having an effect, the planets would continue in a straight inertial line with inertia from the point at which gravity stopped.
The object would crash into the planet.
Planets would most likely drift until they were attracted to the next largest center of gravity, say Jupiter. But without a centre of gravity, the planets would have nothing to hold themselves in a fixed point and would simply drift.
They would fly randomly through space.
The orbits of the planets would all be much larger if the sun had less gravity. They might even just fly off free.
The planets will get too hot than melt/explode.
The planets would no longer follow their orbital paths around the Sun. They would move away from the Sun and travel in straight lines.
The object would crash into the planet.
The planets would fly off into space and lose their moons and atmospheres. The sun would explode from its enormous interior pressure.
Without the Sun's gravity and heat they would all go off in straight lines and freeze up solid.
Yes, gravity affects your weight on different planets. Weight is a measure of the gravitational force acting on an object, so on planets with stronger gravity, you would weigh more, and on planets with weaker gravity, you would weigh less compared to your weight on Earth.