What are some applications outside of mathematics for algebraic geometry? Are there any results from algebraic geometry that have led to an interesting "real world" application? 
 A: The following slideshow gives an explanation of how algebraic geometry can be used in phylogenetics.
See also this post of Charles Siegel on Rigorous Trivialties. This is not an area I've looked at in much detail at all, but it appears that the idea is to use a graph to model evolutionary processes, and such that the "transition function" for these processes is given by a polynomial map.  In particular, it'd be of interest to look at the potential outcomes, namely the image of the transition function; that corresponds to the image of a polynomial map (which is not necessarily an algebraic variety, but it is a constructible set, so not that badly behaved either).  (In practice, though, it seems that one studies the closure, which is a legitimate algebraic set.)
A: Broadly speaking, algebraic geometry is used a lot in some areas of robotics and mechanical engineering. Real algebraic geometry, for example, is important to the development of CAD systems (think NURBS, computing intersections of primitives, etc.) And AG comes up in robotics when it is important to figure out, say, what motions a robotic arm in a given configuration is capable of, or to construct some kind of linkage that draws a prescribed curve.
Something specific in that vein: Kempe's Universality Theorem gives that any bounded algebraic curve in $\mathbb{R}^2$ is the locus of some linkage. The "locus of a linkage" being the path drawn out by all the vertices of a graph, where the edge lengths are all specified and one or more vertices remains still.
Interestingly, Kempe's orginal proof of the theorem was flawed, and more recent proofs have been more involved. However, Timothy Abbott's MIT masters thesis gives a simpler proof that gives a working linkage for a given curve, and makes for interesting reading concerning the problem in general.
Edit: The NURBS connection is, in part, that can construct a B-spline that approximates a given real algebraic curve, which is crucial in displaying intersection curves, for example. See here for more details (I'm afraid I don't know many on this.)
