Well, that certainly is a pickle.
Algebraic geometry is one of the broadest and most important field in mathematics. It has deep connections with almost every area of mathematics, and even theoretical physics (through complex geometry). Number theory, complex geometry, algebraic topology... are all fields that interract very positively with algebraic geometry. So that is already a good reason.
Personally I've been drawn to algebraic geometry through number theory.
And, perhaps an even better reason, is that it is indeed a very beautiful area of mathematics.
But i guess that's not the kind of answers you're looking for.
There are of course very important open questions in the field, but they will be a little difficult to grasp by a non expert. So let's try a more historical approach.
People wanted to study algebraic curves and surfaces in the plane or space. One the first questions to be tackled was a generalisation of d'Alembert-Gauss theorem, that every non constant polynomial will have a root over the complex numbers. Stated in geometric terms it states that a curve defined by the equation y=f(x) where f is a polynomial will generically intersect a line deg(f) times.
So people wanted to generalise this result to arbitrary curves and surfaces.
This led to what is now called intersection theory. How can we properly define the number of intersection of two curves? Can we do it in such a way that two curves will intersect in deg(f)deg(g) points (where the curves are defined by f and g)?
For instance, the intersection of a line and circle tangent to it should count as 2 even though they intersect at a single point, since that if you slightly move the line, you will get 2 distinct points.
This is a very rich and active subfield, and is strongly connected to things like the topology of algebraic varieties, and such things coming from algebraic topology.
There was also enumerative geometry (in fact it is now a part of intersection theory), people wanted to count different type of objects. For instance, on a (non trivial) cubic surface there are exactly 27 lines. Those results have (for long) seem of no particular interrest to me, until i actually looked at the proof, and it is mesmerizing! The amazing point is not that there is 27 lines on a cubic surface, but that we can actually compute such a number (and pretty easlity too).
In the same spirit, people wanted to classify objects. How many genuinely different curves could we build up etc?
You should read this post on MSE.
http://math.stackexchange.com/questions/255063/why-study-algebraic-geometry
Especially the answer from Javier Alavarez (the first answer) which is very complete and i guess accessible to a non expert.