Noise Characterization
There are 2 properties which are important to know about your noise:


Its Self Correlation
What usually called Auto Correlation. This will tell you the relation between 2 noise samples which are apart (On time) from each other (I'm assuming the signal is stationary, namely its properties doesn't change over time).
Best thing to hope for is White Noise.
Its Distribution
Sometimes, in order to be able to do estimations of the data one should also know the distribution of the noise. The most wide spread model is Gaussian Noise as you wrote. Yet on Image Processing it could be Poisson Noise.


If you have access to "Pure" noise samples analyzing those properties of the noise is really easy.
For the first you just calculate the Auto Correlation Function.
For testing Gaussianity / Normality Test (Reference - Tests of Gaussianity) things gets a little tricky.
As you can see in the Wikipedia Page of Normality there are many tests.
Though poor man's (Yet effective) test would be just looking at the Empiric Histogram and compare it to the model (Curve of Normal PDF based on the data Mean and Variance).