What does $\mathbb Z[x]$ mean?

I’ve seen usage of the notation $$\mathbb Z[x]$$ before, where $$x$$ is some non-integer. For example in the Wikipedia article on the Gaussian integers it says that the set of Gaussian integers can be written as $$\mathbb Z[i]$$. But what does it actually mean?

I suspect that $$\mathbb Z[x]$$ refers to the set of numbers of the form $$a+bx$$, where $$a,b\in\mathbb Z$$, but I’m not sure.

$$\mathbb Z[x]$$ is the ring of all polynomials over the variable $$x$$ with integer coefficients. So this includes $$a+bx$$ (for all $$a,b \in \mathbb Z$$), but also things like $$x^2$$, and the other higher order polynomials.
Whereas $$\mathbb Z[i]$$ is what we get when we evaluate the polynomials at $$i$$. That is, we take an integer polynomial $$f(x)$$ and plug in $$i$$ for $$x$$ to get $$f(i)$$, a complex number which we can always write in the form $$a +bi$$ for some integers $$a,b$$.