One can do Euclidean geometry as a completely formal game of symbolic logic. Euclid's axioms are almost sufficient for this, except that they lack formal support for some "obvious" continuity properties such as
Given a circle with center $C$, and a point $A$ such that $|CA|$ is less than the radius of the circle. Then any straight line through $A$ intersects the circle.
David Hilbert, one of the pioneers of the formalist viewpoint, developed an axiomatic system that closes these gaps and allows any Euclidean theorem about a finite number of lines, circles, and points to be proved completely formally. One can work in this system without any reference to arithmetic or set theory, considering Hilbert's geometry axioms as an alternative to, say, ZFC as one's formal basis for one's reasoning.
(Edit: Oops, my history was slightly wrong. Hilbert's axiomatic system was not completely formal. Alfred Tarski later developed a completely formal system; what I say here about Hilbert rightfully ought to read Tarski instead.)
Granted, when one does that one doesn't necessarily think of lines and points as "really existing" in some Platonic sense -- after all, the basic idea of formalism is that no mathematical objects "really exist" and it's all just symbols on the blackboard that we play a parlor game of formal proofs with. But that does not mean that it is necessary, or even desirable, to consider points to be coordinate tuples while playing the game. Indeed, many mathematicians would probably agree that the points and lines of Hilbert's geometric axioms exist "in and of themselves" to at least the same extent that the sets of ZFC (or, for example, the real numbers) exist "in and of themselves".
There are some additional points about this state of things that have the potential to cause confusion, but do not really change the basic facts:
When I speak of Hilbert's axioms as an "alternative" to ZFC, I don't mean that they can be used as a foundation for all of mathematics they way ZFC is -- because they have not been designed to fill that role. I mean merely that they occupy the same ontological position in terms of which concepts one needs to already be familiar with in order to work with them. Perhaps "parallel" might be a better word than "alternative".
The rules of what consists valid formal proofs (in ZFC or geometry) are ultimately defined informally. One may construct a formal model of the rules, but that just punts the informality to the next metalevel, because reasoning about the formal model itself needs some sort of foundation.
When one does formalize the rules of formal proofs, one often does that in a set-theoretic setting. However, this doesn't mean that a different theory such as Hilbert's geometry "depends on" set theory in a fundamental way. Remember that the set theory we use to formalize logic can itself be considered a formal theory, and at some point we have to stop and be satisfied with an informal notion of proof (it cannot be turtles all the way down). And there is no good reason why there has to be a set-theoretic metalayer below the geometry before we reach the inevitable point of no further formalization.
This does not mean that formalizing the rules of formal geometric proof is a pointless exercise. Doing so can tell us things about the axiomatic system that cannot be proved within the formal system itself. In particular, if we formalize the axiomatic system within set theory, we get access to the very strong machinery of model theory to prove facts about the axiomatic system. This is where numbers and coordinates enters the picture, as described in Qiaochu's answer. Using these techniques, one can prove [as a theorem of set theory] that every formal statement about a finite number of lines, points and circles can be either proved or disproved using Hilbert's axioms.
Hilbert's system does not allow one to speak about indeterminate numbers of points in a single statements -- so one cannot prove general theorems about, say, arbitrary polygons. This is by design. The kind of reasoning usually accepted for informal proofs about arbitrary polygons is so varied that formalizing it would probably just end up being a more cumbersome way to do set theory, and there's not much fun in that.
The Kodaira embedding theorem may be of interest. "It says (which) precisely complex manifolds are defined by homogeneous polynomials" c.f. http://en.wikipedia.org/wiki/Kodaira_embedding_theorem
Also the book Griffiths & Harris "Principles of Algebraic Geometry" does algebraic geometry from the complex viewpoint.
Best Answer
Your question on mental abilities is of course too vague to admit a definitive answer, but I'll try to give some reflections on the subject.
1) Essentially, I strongly believe that the differences in abilities necessary to tackle the different branches of mathematics are vastly exaggerated.
In my experience good mathematicians are good at any subject.
The difference between their choices results from mathematics having become so vast that it is very difficult or impossible to have expertise in several subjects, unless you are Serre or Tao.
But my conviction, formed by introspection and anecdotal evidence, is that the subject mathematicians end up with very much depends on chance: books found in a library when 16 years old, teachers had in high-school or university, admired friends,...
To be quite honest, some subjects like combinatorics seem to require special gifts and be a little isolated, but even that is changing: I'm thinking of combinatorists like Stanley who use quite sophisticated "mainstream" mathematics, commutative algebra for example.
2) As for algebraic geometry, it certainly requires no special gifts.
Its origin is Descartes's (and Fermat's) fantastic invention of coordinate geometry, which allows one to solve difficult geometric problems by algebra, in an essentially purely mechanical way (which by the way Jean-Jacques Rousseau didn't like: read the extract from his Confessions in the epigraph to Fulton's Algebraic Curves, page iii)
In the 19-th century and in the 20-th century up to about the 1960's hard algebraic geometry was taught (under the name "analytic geometry") in high schools and a Swiss friend of mine showed me problems he solved when 17 years old, which would baffle most Ph.D holders in algebraic geometry nowadays.
The problem is that, in order in particular to solve quite classical problems and also for arithmetic reasons, algebraic geometers like van der Waerden, Zariski, Weil, Serre, Grothendieck,... had to introduce quite sophisticated machinery, culminating in the notion of scheme.
The unfortunate consequence of those developments is that too many introductory courses spend a semester (say) setting up this wonderful modern machinery and have no time left for showing how to apply it to concrete problems.
The good news is that an antidote to this state of affairs exists: it is called math.stackexchange !
I am amazed at the quality, concreteness and pertinence of many questions and answers relating to algebraic geometry here, and I can only advise you to become a frequent user of our site: just use the tag [(algebraic-geometry)] and start reading, pen in hand!