Gravity in Space: Why is Zero Gravity?

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Why there is zero gravity in space? Shouldn't there always be a resultant gravity in every point in the space from all the stars/planets in the universe?
 
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There is. Far from a massive body, it may be very less.

What force do you think makes the Earth go round the Sun or the moon round the earth?

There is zero gravity wrt to a frame in free fall. For example, in the satellites orbiting the earth, we see that effectively there is no gravity. That is because there is no relative force between the objects inside and the satellite (except for very minute tidal forces), because both are moving at the same rate.

Look up a recent thread which explains this in more detail.