The mass of a body describes how much matter is contained in it. It has constant value that does not depend on where the body is and where the measurement of the mass is being made. Mass is expressed in the SI units kilogram (kg). Weight on the other hand is a force by which an object is pulled towards another due to the gravitational force of attraction between them. The weight of a body is expressed in terms of newtons (N) which is the SI unit for force.
Astronauts in space are weightless; but the reason behind this is not that there is no force of attraction between the Earth and the astronaut. To illustrate the fact, the Moon orbits the Earth due to the gravitational acceleration caused due to the Earth and astronauts have not gone to distances greater than the distance of the Moon from the Earth.
The weightlessness of astronauts is due to the fact that they fall freely towards the Earth as they orbit it in their space craft. If a body is placed on a weighing scale that is falling freely towards the Earth, the scale measures a weight of the body equal to zero. A similar case applies to astronauts. Weightlessness can be experienced by anyone, though for a short duration of time, in many ways, for example in a roller coaster that is moving down at 9.8 m/s^2 or during a bungee jump.
Note also that in stating that astronauts in orbit are weightless, we are neglecting the tiny accelerations, including those due to tidal forces, that affect objects in orbiting spacecraft. Because of these small accelerations, NASA and many space scientists have taken to referring to the conditions in orbiting spacecraft as microgravity, rather than weightlessness. In our opinion, the term microgravity is a poor one for students and tends to feed the common misconception that gravity is absent in space—when, in fact, the acceleration of gravity is only a few percent smaller in low-Earth orbit than on the ground. Perhaps a better term for the conditions in orbit would be microacceleration, but we feel it is pedagogically more useful to simply neglect the small accelerations and refer to the conditions as weightlessness due to free-fall. If you want to be truly accurate, you might refer to the conditions as near-weightlessness and explain why small accelerations still are present.