Yes, there is gravity in space but it is very different from the gravity on earth. Gravity is the force through which an obect can pull other objects towards its center. Every object in space has some sort of gravitational pull, so depending where you are in space the gravitational pull will vary.
The closer an object is to a large object in space the greater the gravitational pull. Astronauts on the moon feel less gravity because they are further away from the earth's gravitational pull. Even the moon has gravity however, so astronauts off the moon feel less gravity than those on the moon.
