Gravity is one of the 4 fundamental fundamental forces and it is the force that attracts objects with mass to each other. Everything has gravity but we will only feel the effects of really big things like planets and stars.
Gravity is a force like the force of a magnet, it is not really produced- its just there! There is a lot of research going on to find out how and why it is there. This is a topic that has kept some of the best scientists busy for centuries- Galileo, Newton and Einstein all worked on this problem.
Any object that has mass will be able to exert a gravitational pull on another object. Every planet will therefore exert a gravitational attraction on other bodies. Gravity is one of the fundamental laws of the universe so it’s pretty inescapable!
There may be no gravity in space but there is a gravitational force between the planets. The satellites of NASA stay in orbit but that does not mean that they stay ‘put’. They move, in a line (like moon orbiting around the earth) and similarly they do have a gravitational force. (This is the answer to your second question in comment section, hope it helps!)