What is zero gravity?
Short answer: True zero gravity doesn't actually exist at all. Every single place in the universe experiences some amount of gravity, even if it's too tiny to measure. Things in space are weightless and float around because they are constantly falling through space. It's the same as being in a plane that's diving straight down—as long as you keep falling, you have no weight, so you float.
Long answer: When scientists and engineers use the term "zero gravity" they don't mean it literally. Many avoid the term altogether, instead using more accurate words such as weightlessness or micro-gravity.
Gravity is caused by the presence of mass, i.e. matter, the stuff that things are made of. Wherever you have a bunch of matter, such as a planet like Earth, you have gravity. Although we don't understand exactly how mass causes gravity, we do know about the effect it has on the space around it. We know that the strength of gravity drops the farther you get from the mass that is causing it, but it never quite drops to zero. This was realized by Sir Isaac Newton, and shown in his law of gravity that states:
"The gravitational attraction force between two point masses is directly proportional to the product of their masses and inversely proportional to the square of their separation distance."
This means that gravitational attraction falls to extremely low levels but never absolutely zero. Therefore, there is at least some effect of gravity throughout the entire universe.
Close to home, we can see that gravity extends throughout our Solar System. Gravity is what holds the Moon in orbit around Earth, and Earth in orbit around the Sun. Farther out, we can see gravitational effects on larger objects and distances, even between galaxies and clusters of galaxies.
The real reason that things are weightless in space is that they are in constant free-fall. It doesn't matter which direction they are travelling, as long as they continue to "fall" smoothly through space without anything stopping them, they will not experience the force of weight.
You only feel weight when something keeps you from falling. On Earth, that's usually the ground. However you could also feel gravity by being in a plane, as long as the plane isn't falling. If it does fall, you become weightless.