Gravity on the International Space Station
The effective gravity inside the ISS is very close to zero, because the station is in free fall. The effective gravity is a combination of gravity and acceleration. (I don't know that "effective gravity" is a commonly used phrase, but it seems to me to be applicable here.)
If you're standing on the surface of the Earth, you feel gravity (1g, 9.8 m/s2) because you're not in free fall. Your feet press down against the ground, and the ground presses up against your feet.
Inside the ISS, there's a downward gravitational pull of about 0.89g, but the station itself is simultaneously accelerating downward at 0.89g -- because of the gravitational pull. Everyone and everything inside the station experiences the same gravity and acceleration, and the sum is close to zero.
Imagine taking the ISS and putting it a mile above the Earth's surface. It would experience about the same 1.0g gravity you have standing on the surface, but in addition the station would accelerate downward at 1.0g (ignoring air resistance). Again, you'll have free fall inside the station, since everything inside it experiences the same gravity and acceleration (at least until it hits the ground).
The big difference, of course, is that the ISS never hits the ground. Its horizontal speed means that by the time it's fallen, say, 1 meter, the ground is 1 meter farther down, because the Earth's surface is curved. In effect, the station is perpetually falling, but never getting any closer to the ground. That's what an orbit is. (As Douglas Adams said, the secret of flying is to throw yourself at the ground and miss.)
But it's not quite that simple. There's still a little bit of atmosphere even at the height at which the ISS orbits, and that causes some drag. Every now and then they have to re-boost the station, using rockets. During a re-boost, the station isn't in free fall. The result is, in effect, a very small "gravitational" pull inside the station -- which you can see in a fascinating NASA video about reboosting the station.
The astronauts are just floating around because they are each in orbit. Because they are at the same altitude as the ISS they are in the same orbit and moving at the same speed as it and so appear weightless.
A simpler way to put this is as follows: when you rotate an object tied to a string there is a centrifugal force pulling the object away from the centre of rotation. This is called centrifugal force. Gravity pulls the object toward the centre of gravity which is also the centre of rotation. Gravity acts like a string. When the centrifugal force is equal to the gravitational force the object is in a stable orbit. The object and all within it are weightless as the two forces cancel out.