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What is gravity like in space?
Short answer
It's still there, and it's strong. At the 400 km altitude of the International Space Station, Earth's gravity retains close to 90% of its surface value. What astronauts experience is microgravity: they fall with the station at the same speed, which is why they feel no weight. The residual accelerations on board are millionths of g.

Microgravity, not zero gravity
The term used on board is microgravity, and it's well chosen. In low Earth orbit, the planet still pulls with almost all of its force: at 400 kilometers of altitude, the altitude of the International Space Station, gravitational acceleration is roughly 90% of that at ground level. No one has escaped gravity; the entire station is falling, and its occupants fall with it. Since the floor falls at the same rate, it doesn't push back against their feet, and without that contact force, the sensation of weight disappears.
What's left isn't exactly zero. The station is subject to tiny disturbances: drag from residual air, the difference in pull between one end of the module and the other, the crew's movements, the fans. Added together they produce accelerations on the order of millionths of Earth's gravity, hence the prefix "micro." The most delicate experiments are mounted on isolating supports to reduce even that residual effect.
Beyond Earth orbit
In deep space the situation changes in nature. Gravity isn't zero, but it's spread among distant bodies and its net effect is tiny; a spacecraft cruising toward Mars is mainly subject to the Sun's pull, which curves its trajectory without anyone on board noticing, because once again everything is falling together. Near a planet or a moon, weight reappears: 1.62 meters per second squared on the Moon, a little over a third of g on Mars.
What it does to the body
Without weight, the skeleton and muscles stop working and degrade. Fluids shift toward the head, which swells the face and puts pressure on the optic nerve, with documented vision changes during long stays. The vestibular system is disoriented during the first few days. The daily counterweight is exercise: about two hours on a treadmill, a bike, and a resistance device that replaces weights, which are useless up there.
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