Gravity is all around us. True, the earth's gravitational attraction lessens as you go further away from it. However, it takes a long time to disappear. Gravity, on the other hand, never totally vanishes. When you get close to another huge body, such as the moon, Mars, or the sun, its gravity overpowers the earth. Only then can the earth's gravity be ignored. Because gravity exists everywhere in space, objects are always falling towards the planet, the sun, and the center. When objects appear to be floating in space without gravity, they are actually falling for two reasons.
To begin with, space is vast and relatively empty in comparison to Earth. When you jump off a bridge, you can tell you're falling because the air is whooshing up, the mountains are shooting up, the water is rapidly approaching, and you feel yourself hitting the water. There is little air to feel whooshing past you as you fall because space is relatively empty, and there are no landmarks to signal you are moving. Because space is so vast, falling through space can take anywhere from hours to years before you reach the surface of a planet, as opposed to the seconds it takes to jump off a bridge.
The second explanation for the lack of apparent gravity in space is that things tend to orbit planets rather than collide with them. Orbiting simply means that an object falls towards a planet and misses it repeatedly owing to gravity. Planets are extremely difficult to impact since space is so vast and planets are so little in contrast.