Guage invariance is simply a redundancy in the description of a physical system. I.e. we can choose from an infinite number of vector potentials in E&M
For example, an ifinite number of vector potentials can describe electromagnetism by the transformation below
$A -> A^\mu + \Delta^\mu \alpha $
(The \Delta above should be del - if someone could tell me how to write del i would be much obliged)
Choosing a specific guage (guage fixing) can make solving a physical problem much easier than it would be if you did not fix a guage.
Normally one chooses the coulomb guage; $\Delta \cdot A = 0$