Interesting answer: I will read more carefully later. Also, I suspect I made a mistake: the fiber product, as a scheme over $k$, should really sit inside the product (X\timesY), NOT inside $X\timesY\timesZ$ as it's written in my question. Should I edit?

Ah, can assume $k$ is a field if you want: just to simplify matters, but perhaps it's not relevant.

@Mike: Yes, you can assume familiarity with the tensor product thing. The answer is "yes" also to your second question, in some sense. As far as I understand, the euristic interpretation of fiber product is: "construct a space $P$ over $Z$ such that its fiber $P_z$ over the point $z\inZ$ is just the absolute (over $Spec(k)=$ the residue field at $z$) product of the fibers: $X_z\timesY_z$". Does it sound correct?

And in this case $X$ is even smooth (though not projective).

Something like: $(x,y)\mapsto(y^2+x,y^3)$

I don't think it's true: $\mathbb{A}^2$ is irreducible, but consider the map from $\mathbb{A}^2$ to itself that sends each vertical line into a "translated" cuspidal cubic...

By the way, I don't know if those twisted locally constant sheaves (local systems?) are the right candidates to be a structure sheaf for some geometrical object. Not if I want "regular functions" to correspond to morphism to the "usual" $\mathbb{A}^{1}$, I'd guess.

@fpqc again: I'm aware of the "schemes-->algebraic spaces-->DM stacks---> Artin stacks" generalizations. But here it seems to me that we're generalizing (if possible) in another direction: about things that do not look like Spec(A) even zariski-, étale-, etc- locally.

(continued) Uhm, in fact one should add some topological requirements on the subsheaf of set-theoretical funcions I mentioned above. In that context, as far as I remember, you do have a well defined notion of quotient (which of course in general will fail to be a variety), and will have even the definition of an induced structure of the same kind on any subset.

@fpqc: Actually, there exists a precise definition of "quotient ringed space", at least if the structure sheaf is a subsheaf of the (sheaf of rings of) set-theoretical functions with values in a given field. I came across this defn. while attending an introductory course of a.g. (in which alg. varieties, using an approach "à la Serre", were considered a subcategory of these ringed spaces).

Does it tell you anything about the proj.form. in the Chow group case?

(btw: I haven't downvoted this question)

In my country we study epsilon-delta definition of continuity (and all elementary theorems about calculus such as mean-value thm and the fundamental thm of calculus with rigorous proofs) at the last year of high school...