If $\mathbf{x} \in \mathbb{R}^n$, then we would have $x_1, x_2, ..., x_n \in \mathbb{R}$, right? This is commonly known as $n$-dimensional space.
My question is, could we also have such a thing as $\mathbf{x} \in \mathbb{Z}^n$ such that $x_1, x_2, ..., x_n \in \mathbb{Z}$? In other words, could we also have $n$-dimensional integer space? Or would this simply be equivalent to $\{ \mathbf{x} \in \mathbb{R}^n | x_1, x_2, ..., x_n \in \mathbb{Z} \}$?
I would greatly appreciate it if people could please take the time to clarify this.
Yes, we talk about such things all the time: its called the notation for the Cartesian power as you are just using the Cartesian product $n$ times with a single set.
Yes, you can always form the set, but using "dimensional" may not always be appropriate. Outside of vector spaces, which are direct products of copies of fields, the number of factors in the Cartesian power may not be a relevant quantity. And sometimes it is important and just has a different name. For example, if you're considering $R^n$ as a module over itself, we usually say "$R^n$ is a free module of rank $n$" instead of "$R^n$ is $n$ dimensional."
The question is "what do you mean by equivalent?" The answer should something like "I mean up to isomorphism in some category."
In that case, the answer to this would be "yes": just as we can identify $\mathbb Z$ as a subset of $\mathbb R$ (there is an inclusion map from $\mathbb Z\to \mathbb R$) we can identify $\mathbb Z^n\subset\mathbb R^n$ via an inclusion map. This inclusion works for all of the categories you probably have in mind, namely the category of sets, abelian groups, or rings.