Greg's example from Hartshorne is actually a special case of a more general situation. Under any dominant map of affine varieties, the inverse image of the generic point is the scheme associated to a finitely generated domain over the function field of the target. Hence by Noether's normalization, this inverse image scheme is a finite cover of an affine space over that function field. It follows that over an open set U of the target, the map factors as a finite cover of the projection of U x k^n --> U, where n is the transcendence degree of the field extension defined by the original map. In Hartshorne's exercise of course n = 0.
This is the argument for the structure of a dominant morphism in Mumford's red book, I.8, proof of theorem 3.
In my experience the word yoga has been explained to mean "yoke" or "union", from the Sanscrit, and refers to any practice meant to help achieve oneness, or perhaps understanding, as the unknown touched upon above. But my impression is that practicing yoga is more spiritual than intellectual.
This is going to be perhaps vague, but I'l try to write down the idea.
I've been told that in doing scheme theory, most of the time what is studied is not a scheme but a morphism of schemes $f:X\rightarrow S$. The studied of such a maps can be enriched allowing a chance of "base" $S$. Such a scheme $S$ can be crazy, but I'll impose finite properties to the map $f$ to keep everything under control.
Now, as you say, if we have two $S$-schemes $f:X\rightarrow S$ and $Y\rightarrow S$ then we can consider the fiber product of them (which is a categorical product of $X$ and $Y$ over $S$). Such a process can be though of as replacing the base $S$ by the scheme $Y$. In doing so, the new map $f_Y:X_Y\rightarrow Y$ (standing for $p_2:X\times_{S}Y\rightarrow Y$) may be easier to work with than the map itself $f$. This very construction is generalizing the idea of "extending scalars".
Here is a sort of example of a product described above. Given the point $s\in S$ and its residue field $k(s)$ of the local ring $\mathcal{O}_s$ at that point. Then it is well known that $X_s=X\times_S Spec(k(s))$ has an underlying space the "fiber" $f^{-1}(s)$ (as long as $f$ is "finite"). This space can be considered as an algebraic variety over the field $k(s)$. This way $S$ is the scheme parameterizing the varieties $X_s$ some of which a priori may have different ground fields $k(s)$.
On the other hand, back to the fiber product, in considering a variety $X$ over the field $k$ (scheme of finite type over $k$), we can extend the scalars from $k$ to a field extension $K$. In doing so, we may think of such a variety now over a field extension $K$. Here is where the product comes into play. Vaguely enough, this is saying that if you have an equation and you have solved it over the field $k$ now you might ask yourself if you'd be able to solved it again but over a field extension $K$, here you want to perform an extension of scalars and the ideas written above may help out.
Needless to say that we can consider as well $X\times_k Spec(\overline{k})$ and due to the fact that field $k$ may not have been algebraically closed, such a fiber product can be handful.
Now, to get intuition, let's take a look at an example: $$\pi:Spec \mathbb{Z}[i]\rightarrow Spec(\mathbb{Z})$$
Let $X=Spec\mathbb{Z}[i]$ and notice that the fiber $$X_p=X\times_{\mathbb{Z}}\mathbb{F}_p=Spec(\mathbb{Z}[i]\otimes\mathbb{F}_p)$$
Such a fiber is going to have cardinality 2 if $p\equiv 1 mod(4)$ and cardinality 1 if $p\equiv 3 mod(4)$ (this is the fact that $p=a^2+b^2$ where $a,b\in \mathbb{Z}$ if $p\equiv 1mod 4$). These fibers are $X$ reduce mod $p$. Notice that at the generic point we have $X_0=\mathbb{Q}[i]$.
Now the $T-points$ in $X$ where $T=\mathbb{F}_p[i]$, and all that story give rise to the functor which represents the scheme $X$. If I am not wrong, the info that carries such a functor is nothing but that of the fibers of the map $\pi$. So naively it is like having $X$ fibers wise.
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My favorite view on generic points is found in Mumford's book: Complex Projective Varieties, on page 2. It goes as roughly as follows:
Definition: Let $k \subset \mathbb{C}$ be a subfield of the complex numbers and $V$ an affine complex variety. A point $x \in V$ is $k$-generic if every polynomial with values in $k$ that vanishes on $x$, vanishes on all of $V$.
Proposition: If $\mathbb{C}/k$ has infinite transcendental degree, then every variety $V$ has a $k$-generic point.
Proof: Extend $k$ by all coefficients of a finite set of equations for $V$. Note that $\mathbb{C}/k$ has still infinite transcendental degree. But now $V$ becomes a variety over $k$ in a canonical way. The function field $L$ of $V$ is an $k$-extension of finite transcendental degree, and therefore can be embedded into $\mathbb{C}$. The images of the coordinate function $X_i \in L$ in $\mathbb{C}$ give the coordinates of a $k$-generic point.
For the relation to the generic point $\eta \in V$ from scheme theory note the following:
From the abstract perspective the function field $L=K(\eta)$ is as good as $K(x)$, if not better, because it does not depend on choices. Moreover, all $k$-linear algebraic operations cant tell a difference between $\eta$ and $x$.