What is "noise" in observed data?
When you have sensors, the values you receive change even if the signal that was recorded didn't change. This is one example of noise.
When you have a model of the world, it abstracts from the real relationships by simplifying things which are not too important. To take into account for the simplification, you model the error as noise (e.g. in a Kalman filter).
But noise sources can be anything. For example, in an image classification problem, data compression can distort an image. Images can have different resolution; low resolution signals are harder to classify than high resolution figures. Aliasing effects can also distort images.

And what is additive noise?

Suppose your system equation is
$$z = H \cdot x$$
where $z \in \mathbb{R}^{n_m}$ is the observation, $x \in \mathbb{R}^{n_x}$ is the state you're interested in and $H \in \mathbb{R}^{n_m \cdot n_x}$ is a transformation matrix. Then the noise could interact with your system in any way. But most of the time it is logical and practical that the noise is additive, meaning your model is
$$z = H \cdot x + r$$
where $r$ is sampled from a random variable of any distribution.

What is (additive) Gaussian noise?

$$r \sim \mathcal{N}(\mu, \sigma^2)$$
See normal distribution