All objects with mass, meaning all objects, generate a gravitational field. In doing so, they are attracted to other objects by a gravitational force between any two objects that can be calculated by the well known equation $F = Gm_1m_2/r^2$. This force of gravitational attraction exists when you are in free fall and when you are standing on the surface of the earth.
Weight is normally determined when you step on a scale, depress the springs in that scale, and take a measurement off of a dial of some sort. When you step on the scale, you are pushing on it due to the force of gravitational attraction between you and the earth. The opposing force from the scale, due to Newton's third law, is the normal force that the scale puts back on the bottom of your feet, and that force is in the upwards direction. It is actually the normal force from the scale that is displayed on the scale's dial, and this reading is what you call your weight.