Easy to state applications of dimension theory in algebraic geometry Dimension theory is quite a sophisticated topic (at least for me), it is fully settled in Shafarevich's book on the first 100 pages. 
Shafarevich gives two nice applications of the theory. 1) A proof of Tzen's theorem 2) A proof of existence of a line on a cubic surface in $\mathbb P^3$. 
Both applications are geometric statements. I would like to learn about some other applications that are easy to state. I am asking this partially because of an introductory course in algebraic geometry that I am teaching. I would like to give some motivation for developing the dimension theory.
 A: Here is a cool idea which gets used a lot. Suppose that a connected algebraic group $G$ acts on a variety $X$. One can prove that the orbits are locally closed using Chevalley's theorem on constructible sets. Moreover, the boundary of an orbit must be a union of orbits. This implies that the orbits of minimal dimension are closed. The most important application of this is the following:

When a connected solvable group acts on a projective variety, there is at least one fixed point.

A: This isn't a very direct answer, but one of the things I use all the time in algebraic geometry as opposed to non-algebraic is that if $X$ is dense in $Y$, then $\dim(Y\setminus X) < \dim X$. The topologist's sine curve is a counterexample to this in non-algebraic geometry.
A: I've taught an introductory course recently and regarding students' reaction, Bertini was the best application ever.
A: It's hard to know where to start, since dimension theory is used everywhere. But here is another nice simple geometric application: Any smooth curve in projective space is isomorphic to a curve in $\mathbb{P}^3$. To prove this, start with a curve $C\subset\mathbb{P^n}$, with $n\ge 4$. Observe by computing dimensions that the set of secant and tangent lines won't sweep out of all of $\mathbb{P}^n$. So a general projection will map $C$ to an isomorphic curve in $\mathbb{P}^{n-1}$.
