Why study Algebraic Geometry? I'm going to start self-stydying algebraic geometry very soon. So, my question is why do mathematicians study algebraic geometry? What are the types of problems in which algebraic geometers are interested in? And what are some of the most beautiful theorems in algebraic geometry?
NEW ADDITION: a big list of freely available online courses on algebraic geometry, from introduction to advanced topics, has been compiled in this other answer. And a digression on motivation for studying the subject along with a self-learning guide of books is in this new answer.

There are other similar questions, above all asking for references for self-studying, whose answers may be helpful:


(Undergraduate) Algebraic Geometry Textbook Recomendations.
Reference for Algebraic Geometry.
Best Algebraic Geometry text book? (other than Hartshorne).


My personal recommendation is that you start and get your motivation in the following freely available notes. They are extremely instructive, from the very basics of complex algebraic curves up to schemes and intersection theory with Grothendieck-Riemann-Roch, and prove of some of the theorems I mention below. They are excellent for self-study mixing rigor with many pictures! (sadly, something quite unusual among AG references):


Matt Kerr - Lecture Notes Algebraic Geometry III/IV, Washington University in St. Louis.
Andreas Gathmann - Class Notes: Algebraic Geometry, University of Kaiserslautern.


For a powerful, long and abstract course, suitable for self-study, these notes have become famous:


Ravi Vakil - Foundations of Algebraic Geometry, Stanford University.


Also, there are many wonderful lecture videos for complete courses on elementary algebraic geometry, algebraic surfaces and beyond, by the one old master:


Miles Reid - Lecture Courses on Video (WCU project at Sogang University),


where you can really start at a slow pace (following his undergraduate textbook) to get up to the surface classification theorem.



Now, Algebraic Geometry is one of the oldest, deepest, broadest and most active subjects in Mathematics with connections to almost all other branches in either a very direct or subtle way. The main motivation started with Pierre de Fermat and René Descartes who realized that to study geometry one could work with algebraic equations instead of drawings and pictures (which is now fundamental to work with higher dimensional objects, since intuition fails there). The most basic equations one could imagine to start studying were polynomials on the coordinates of your plane or space, or in a number field in general, as they are the most basic constructions from the elementary arithmetic operations. Equations of first order, i.e. linear polynomials, are the straight lines, planes, linear subspaces and hyperplanes. Equations of second order turned out to comprise all the classical conic sections; in fact the conics classification in the affine, Euclidean and projective cases (over the real and complex numbers) is the first actual algebraic geometry problem that every student is introduced to: the classification of all possible canonical forms of polynomials of degree 2 (either under affine transformations or isometries in variables $(x,y)$, or projective transformations in homogeneous variables $[x:y:z]$). Thus the basic plane curves over the real numbers can be studied by the algebraic properties of polynomials. Working over the complex numbers is actually more natural, as it is the algebraic closure of the reals and so it simplifies a lot the study tying together the whole subject, thanks to elementary things like the fundamental theorem of algebra and the Hilbert Nullstellensatz. Besides, working within projective varieties, enlarging our ambient space with the points at infinity, also helps since then we are dealing with topologically compact objects and pathological results disappear, e.g. all curves intersect at least at a point, giving the beautiful Bézout's theorem.

From a purely practical point of view, one has to realize that all other analytic non-polynomial functions can be approximated by polynomials (e.g. by truncating the series), which is actually what calculators and computers do when computing trigonometric functions for example. So when any software plots a transcendental surface (or manifold), it is actually displaying a polynomial approximation (an algebraic variety). So the study of algebraic geometry in the applied and computational sense is fundamental for the rest of geometry.

From a pure mathematics perspective, the case of projective complex algebraic geometry is of central importance. This is because of several results, like Lefschetz's principle by which doing (algebraic) geometry over an algebraically closed field of characteristic $0$ is essentially equivalent to doing it over the complex numbers; furthermore, Chow's theorem guarantees that all projective complex manifolds are actually algebraic, meaning that differential geometry deals with the same objects as algebraic geometry in that case, i.e. complex projective manifolds are given by the zero locus of a finite number of homogeneous polynomials! This was strengthened by Jean-Pierre Serre's GAGA theorems, which unified and equated the study of analytic geometry with algebraic geometry in a very general setting. Besides, in the case of projective complex algebraic curves one is actually working with compact orientable real surfaces (since these always admit a holomorphic structure), therefore unifying the theory of compact Riemann surfaces of complex analysis with the differential geometry of real surfaces, the algebraic topology of 2-manifolds and the algebraic geometry of algebraic curves! Here one finds wonderful relations and deep results like all the consequences of the concept of degree, index and curvature, linking together the milestone theorems of Gauß-Bonnet, Poincaré-Hopf and Riemann-Roch theorem! In fact the principal classification of algebraic curves is given in terms of their genus which is an invariant proved to be the same in the different perspectives: the topological genus of number of doughnut holes, the arithmetic genus of the Hilbert polynomial of the algebraic curve and the geometric genus as the number of independent holomorphic differential 2-forms over the Riemann surface. Analogously, the study of real 4-manifolds in differential geometry and differential topology is of central importance in mathematics per se but also in theoretical and mathematical physics, for example in gauge theory, so the study of complex algebraic surfaces gives results and provides tools. The full birational classification of algebraic surfaces was worked out decades ago in the Kodaira-Enriques theorem and served as a starting point to Mori's minimal model program to birationally classify all higher-dimensional (projective) complex algebraic varieties. A fundamental difference with other types of geometry is the presence of singularities, which play a very important role in algebraic geometry as many of the obstacles are due to them, but the fundamental Hironaka's resolution theorem guarantees that, at least in characteristic zero, varieties always have a smooth birational model. Also the construction and study of moduli spaces of types of geometric objects is a very important topic (e.g. Deligne-Mumford construction), since the space of all such objects is often an algebraic-geometric object itself. 
There are also many interesting problems and results in enumerative geometry and intersection theory, starting from the classic and amazing Cayley-Salmon theorem that all smooth cubic surfaces defined over an algebraic closed field contain exactly 27 straight lines, the Thom-Porteus formula for degeneracy loci, Schubert calculus up to modern quantum cohomology with Kontsevich's and ELSV formulas; Torelli's theorem on the reconstruction of algebraic curves from their Jacobian variety, and finally the cornerstone (Grothendieck)-Hirzebruch-Riemann-Roch theorem computing the number of independent global sections of vector bundles, actually their Euler-Poincaré characteristics, by the intersection numbers of generic zero loci of characteristic classes over the variety.

Besides all this, since the foundational immense work of Alexandre Grothendieck, the subject has got very solid and abstract foundations so powerful to fuse algebraic geometry with number theory, as many were hoping before. Thus, the abstract algebraic geometry of sheaves and schemes plays nowadays a fundamental role in algebraic number theory disguised as arithmetic geometry. Wondeful results in Diophantine geometry like Faltings theorem and Mordell-Weil theorem made use of all these advances, along with the famous proof of Wiles of Fermat's last theorem. The development of abstract algebraic geometry was more or less motivated to solve the remarkable Weil conjectures relating the number of solutions of polynomials over finite number fields to the geometry of the complex variety defined by the same polynomials. For this, tremendous machinery was worked out, like étale cohomology. Also, trying to apply complex geometry constructions to arithmetic has led to Arakelov geometry and the arithmetic Grothendieck-Riemann-Roch among other results.

Related to arithmetic geometry, thanks to schemes, there has emerged a new subject of arithmetic topology, where properties of the prime numbers and algebraic number theory have relationships and dualities with the theory of knots, links and 3-dimensional manifolds! This is a very mysterious and interesting new topic, since knots and links also appear in theoretical physics (e.g. topological quantum field theories). Also, anabelian geometry interestingly has led the way to studies on the relationships between the topological fundamental group of algebraic varieties and the Galois groups of arithmetic number field extensions.

So, mathematicians study algebraic geometry because it is at the core of many subjects, serving as a bridge between seemingly different disciplines: from geometry and topology to complex analysis and number theory. Since in the end, any mathematical subject works within specified algebras, studying the geometry those algebras define is a useful tool and interesting endeavor in itself. In fact, the requirement of being commutative algebras has been dropped since the work of Alain Connes and the whole 'new' subject of noncommmutative geometry has flourished, in analytic and algebraic styles, to try to complete the geometrization of mathematics. On the other hand it attempts to give a quantum counterpart to classical geometries, something of extreme interest in fundamental physics (complex algebraic geometry and noncommutative geometry appear almost necessarily in one way or another in any attempt to unify the fundamental forces with gravity, i.e. quantum field theory with general relativity; even abstract and categorical algebraic geometry play a role in topics like homological mirror symmetry and quantum cohomology, which originated in physics).

Therefore, the kind of problems mathematicians try to solve in algebraic geometry are related to much of everything else, mostly: anything related to the classification (as fine as possible) of algebraic varieties (and schemes, maybe someday), their invariants, singularities, deformations and moduli spaces, intersections, their topology and differential geometry, and framing arithmetic problems in terms of geometry. There are many interesting open problems: 


Birational minimal model program for all varieties, 
Hodge conjecture,
Jacobian conjecture,
Hartshorne's conjecture,
General Griffiths conjecture,
Fujita's conjecture,
Linearization and cancelation conjectures,
Coolidge-Nagata conjecture,
Resolution of singularities in nonzero characteristic,
Grothendieck's standard conjectures on algebraic cycles,
Grothendieck's anabelian section conjecture,
Classification of vector bundles over projective spaces,
Unirationality of moduli spaces of curves,
Unirationality of rationally connected varieties,
Full rigorous formalization of mirror symmetry and quantum cohomology,
Full theory of a universal cohomology and mixed motives (e.g. Voevodsky vanishing conjecture).


In my personal case, I started as a theoretical physicists but switched completely to pure mathematics because of algebraic geometry, and I also began by self-learning. It is a very deep subject with connections to almost everything else, once one has learned enough to realize that. It is also a very demanding field because of the tremendous background one has to master, in commutative and homological algebra for example, before being able to get to the most modern and interesting results. The effort nevertheless pays off! In fact, the route through commutative algebra actually paves the way not only to algebraic geometry but to algebraic number theory and arithmetic geometry. I had a strong background in differential geometry so I arrived at algebraic geometry through complex (Kähler) geometry, and ended up fascinated by even the most abstract incarnations of it.

"Algebraic geometry seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics. In one respect this last point is accurate..." - David Mumford.

So the question could be instead "why not study algebraic geometry!?" I hope this answer motivates you enough to dive into this deep ocean of the mathematical world and to corroborate it yourself. Best luck!
The first chapter of Justin Smith's Introduction to Algebraic Geometry has a nice discussion of Bezout's Theorem and how algebraic geometry approaches geometric problems in general. I think you can download the PDF from the author's web site for free.