In 'gauge symmetry', 'gauge' means 'measure', and symmetry means 'stays the same'. Geometry is the study of the properties of objects that do not change when they move around. An object is symmetric if some motion leaves it looking the same, for instance, rotating an equilateral triangle through sixty degrees. Many physical properties are invariant under translation and rotation. For instance, the results of an experiment are not dependent on where the experimental apparatus is, or in which direction it faces.
Gravity is a gauge theory because its predictions stay the same when measurements are taken from different baselines. For instance, a ball on a staircase has gravitational potential energy. If it moves down a step its loss in energy depends only on the strength of the gravitational field and the height dropped. You can measure its gravitational potential energy from anywhere (earth's surface, another step, Alpha Centauri...) and the difference in energy between the two steps 'stays the same'. This global invariance in the measurement procedure makes gravity a gauge theory, i.e. a field theory showing gauge symmetry.
