What is U(1)-Gauge Symmetry?

It’s the gauge symmetry of quantum electrodynamics (QED).

What it is is a circular symmetry group in the complex plane that leaves a vectors magnitude unchanged. In this case, the ‘vector’ is the electromagnetic gauge field (electromagnetic 4-potential). What this means is that observables of the electromagnetic field - which are derived from the gauge field - are unchanged under the U(1) symmetry transformation. 

When you make appropriate changes to the Schrödinger wave function by incorporating the covariant gauge derivative to compensate for the fact that the standard wave function is not U(1) symmetric, you end up with QED a complete description of the charged matter-field interaction.

The other symmetries incorporated into the standard model are SU(2) and SU(3). These are Lie symmetry groups (special unitary) in 2 and 3 complex dimensions that are the set of all 2 and 3 dimension rotations in the complex space that preserve the magnitude of complex vectors and angles between them. So, SU(2) has a representation that are complex 2x2 matrices that can transform the higher dimensional gauge fields in a way that doesn’t effect observables.