The Question:
What is a ringed space? Specifically, how does one think about them?
Context:
Ringed spaces are important for many fields of mathematics, but for me, I use them in the context of algebraic groups.
Let $X$ be a Zariski topological space and $f\in k[X]$ for a field $k$. Define
$$D_X(f)=D(f)=\{x\in X\mid f(x)\neq 0\}.$$
I struggled with them upon my first encounter, from Springer's, "Linear Algebraic Groups (Second Edition)":
1.4.1. Let $x\in X$. A $k$-valued function $f$ defined in a neighbourhood $U$ of $x$ is called regular in $x$ if there exist $g,h\in k[X]$ and an open neighbourhood $V\subseteq U\cap D(h)$ of $x$ such that $f(y)=g(y)h(y)^{-1}$ for all $y\in V$.
A function $f$ defined on a non-empty open subset $U$ of $X$ is regular if it is regular in all points of $U$. So for each $x\in U$ there exist $g_x,h_x$ with the properties stated above. Denote by $\mathcal{O}_X(U)$ or $\mathcal{O}(U)$ the $k$-algebra of regular functions in $U$. The following properties are obvious:
(A) If $U$ and $V$ are non-empty open subsets and $U\subset V$, restriction defines a $k$-algebra homomorpism $\mathcal O(U)\to \mathcal O(V).$
(B) Let $U=\bigcup_{\alpha\in A}U_\alpha$ be an open covering of the open set $U$. Suppose that for each $\alpha\in A$ we are given $f_\alpha\in \mathcal O(U_\alpha)$ such that if $U_\alpha \cap U_\beta$ is non-empty, $f_\alpha$ and $f_\beta$ restrict to the same element of $\mathcal O(U_\alpha\cap U_\beta)$. Then there is $f\in\mathcal O(U)$ whose restriction to $U_\alpha$ is $f_\alpha$ for all $\alpha\in A$.1.4.2. Sheaves of functions. Now let $X$ be an arbitrary topological space. Suppose that for each non-empty open subset $U$ of $X$, a $k$-algebra of $k$-valued functions $\mathcal O(U)$ is given such that (A) and (B) hold. The function $\mathcal O$ is called a sheaf of $k$-valued functions on $X$. [. . .] A pair $(X,\mathcal O)$ consisting of a topological space and a sheaf of functions is called a ringed space.
Now, that's all well & good, but condition (B) is a little involved; and in coming back to the basic material for a report I'm giving, I'm struggling with it once more.
The Wikipedia article on ringed spaces is somewhat useful, although its definition is a little terse:
A ringed space $(X,\mathcal{O}_X)$ is a topological space $X$ together with a sheaf of rings $\mathcal{O}_X$ on $X$.
It's not clear to me how these are the same as what Springer describes; maybe they're not.
Answers:
An ideal answer would describe the intuition behind ringed spaces at the level of an advanced undergraduate with little to no experience with topology. (My topology has always been a bit weak.)
Please help :)
Let's start with a motivating example. Given any topological space $\mathcal{X}$ (I'm not sure what you mean by "Zariski topological space," but it doesn't matter) and an open set $U$ in $\mathcal{X}$, the "nice" functions $U\rightarrow\mathbb{R}$ will form a ring under pointwise addition and multiplication. Here I'm being a bit vague about what "nice" means; to start with, we could take "nice" to be continuous, but generally we'll want to upgrade to something more powerful like smooth. We may also want to change the codomain from $\mathbb{R}$ to $\mathbb{C}$, or indeed any field $k$. But let's start with the "continuous real-valued functions" version for concreteness.
Now not only does each open $U$ give rise to a ring $F_U$ of "nice" functions, but these rings "fit together" well. Specifically, the assignment $$\mathfrak{S}:U\mapsto F_U$$ is a sheaf. You mention that the definition of a sheaf is rather involved; unfortunately, it really is one of those notions that deserves a careful definition. Luckily, I can point to the specific structure that's at play here in terms of motivating it: whenever $U\subseteq V$ we can restrict a continuous function from $V$ to a continuous function from $U$, and this means we have a canonical ring homomorphism $h_{U,V}:F_U\rightarrow F_V$. The general notion of a sheaf comes from the data $$(\mathfrak{S}, \{h_{U,V}: U\subseteq V\})$$ consisting of not just the assignment of rings of functions but also the canonical homomorphisms fitting them together. One obvious property this has to satisfy is that "restricting twice" is the same as "restricting once," in the sense that $$h_{V,W}h_{U,V}=h_{U,W}$$ (remember that composition notation is garbage); another obvious property is that $h_{U,U}=id_{F_U}$. Unfortunately, it turns out that there is a bit more structure at play, and recognizing the importance of these further properties (locality and gluing) is what takes us from presheaves to sheaves.
Since this is a big chunk of your question, let me describe it in some detail. Let's start with locality. Imagine I have open sets $V,U_1,U_2,U_3$ with $V=U_1\cup U_2\cup U_3$. Then if $f$ is a continuous function on $V$, I can recover $f$ from its restrictions $f_{\vert U_1}, f_{\vert U_2}, f_{\vert U_3}$. (E.g. thinking set-theoretically, $f$ literally is $f_{\vert U_1}\cup f_{\vert U_2}\cup f_{\vert U_3}$; of course this needs tweaking if we're keeping track of codomains of functions, but meh.) Note that the continuity of $f$ and the open-ness of the $U_i$s and $V$ are completely irrelevant here; this is a general property about recovering a function from enough data.
The locality axiom generalizes this phenomenon: in the particular case above, it says that whenever $f_1,f_2\in F(V)$ we have $$f_1=f_2\quad\iff\quad h_{V,U_1}(f_1)=h_{V,U_1}(f_2)\mbox{ and } h_{V,U_2}(f_1)=h_{V,U_2}(f_2)\mbox{ and } h_{V,U_3}(f_1)=h_{V,U_3}(f_2).$$ Think of locality as a kind of extensionality principle: while extensionality lets us infer equality of sets by checking elementwise, locality lets us infer equality of "big" sections by checking restrictionwise.
But we also want to be able to go in the opposite direction: "piecing together" open sets doesn't just let us check whether two functions are equal, it also lets us build new functions out of small pieces. This is what the gluing axiom says: if I have a bunch of "small" open sets $U_i$ ($i\in I$) with $\bigcup_{i\in I}U_i=V$, and I have continuous functions $f_i\in F(U_i)$ which "agree on all overlaps," then I should be able to "piece the $f_i$s together" and get a single $f\in F(V)$. Saying that this $f$ is gotten by piecing the $f_i$s together amounts exactly to saying $$h_{V,U_i}(f)=f_i.$$
Continuing the above analogy, as a set theorist I've found the intuition "locality is to gluing as extensionality is to comprehension" a useful one ("comprehension" is just a fancy term for "set-formation" - if you're most familiar with $\mathsf{ZFC}$ specifically, think separation or replacement). I don't know if anyone else agrees with this.
Now we have a choice of levels of generality. Given a topological space $\mathcal{X}$, we have the notion of a sheaf of rings of real-valued functions on open subsets of $\mathcal{X}$. But we could also just talk about a sheaf of rings on $\mathcal{X}$, and not require that the elements of each $F(U)$ literally be functions on $U$. (We could go even further and talk about sheaves of whateverthehecks on $\mathcal{X}$, but let's stop here for now.)
My understanding is that this is somewhat context-dependent: while the term "ringed space" technically carries the broader definition (so that there is not assumed to be any way to "apply" an element of the ring $F(U)$ to an element of the open set $U$), in many cases (such as the book you mention) it is enough to look at the special case of "function-ringed" spaces, and these are also just called "ringed spaces" in a slight abuse of terminology. But this is where my expertise ends, so someone else should chime in here.