TITLE: Geometric fibers of schemes.
QUESTION [10 upvotes]: I would like to get an understanding of the notion of geometric fibers of scheme morphisms: If $f: X \rightarrow Y$ is a morphism of schemes, then its geometric fiber is defined to be $X \times_{Y} \overline{k(p)}$ for the quotient field $k(p)$ at $p \in Y$. I would like to know, why this is a good choice for the notion of "fiber". Why does one pick such an abstract notion and why does exactly this the right job?
Every help will be appreciated.

REPLY [3 votes]: Jacob has already given a good answer. I just want to add something that convinces me that the definition of geometric fibre is indeed a good one.
Let's fix a field $k$, and consider the concept of smoothness of a variety $X$ over $k$ (or more generally, schemes locally of finite type over $k$).
It's quite natural to call some (closed) point $x \in X$ a smooth point if   $\Omega_{X/k,x}$ if free of rank equal the local dimension of $X$ at $x$. Geometrically, this simply states there are enough independent differentials locally. 
We would like to express this condition in terms of the local ring $O_{X,x}$. In the case $k$ is algebraically closed, this condition is (as is well known) that $O_{X,x}$ is  a regular local ring.
However, in general, the equivalent condition is $O_{X,x}\otimes_k \bar{k}$ is regular. (This means each localization is regular, as in general this ring is only semi-local). Here geometric fibre arises naturally. 
The reason for this is somehow technical, there is a result characterizing regular local rings in terms of Kaehler differentials, see for example Hartshorne Thm II.8.8 (notice that there the assumption that $k$ being perfect is superfluous).
Theorem: 
Let $B$ be a local ring containing a field $k$ isomorphic to its residue field. Assume furthermore that $B$ is a localisation of a finitely generated $k$-algebra. Then $\Omega_{B/k}$ is a free $B$-module of rank equal to dim $B$ if and only if $B$ is a regular local ring.
The first assumption is clearly satisfied for varieties over an algebraically closed field, but fails in general. So a base extension to the algebraic closure arises quite naturally.