In this article, Doris Schattschneider defines what a repeating (or periodic) pattern is. The definition goes as follows:
A periodic pattern in the plane is a design having the following property: there exist a finite region and two linearly independent translations such that the set of all images of the region when acted on by the group generated by these translations produce the original design.
I have a couple questions regarding this definition. I'm clear on what she means by translations and group actions, but what does she mean when she says "the set of all images of the region", is she referring to all the images of all the points of the region?
For example, consider the following periodic pattern (called Study of Regular Division of the Plane with Fish and Birds, due to M.C. Escher and taken from the web)
My intuition says that the region she's demanding is one of the parallelograms 1-2;2-1 (and pardon the notation), the two independent translations are the ones whose vectors are the sides of the parallelogram. Is it correct?
Yes, your intuition is correct. Such region is called 'fundamental region' for the translation action, meaning that its interior contains exactly one element of each orbit. So 'the set of all images' means the set of orbits of the elements of such region under the translation action.
Recall that, given a group $G$ acting on a set $X$, the orbit of an element $x \in X$ is the set $\{g \cdot x : g \in G\}$.
Indeed, the two linearly independent vectors may be the sides of the parallelogram. Note that such pair of vectors, as well as fundamental regions, are not unique.