Let me give a different perspective from countinghaus' nice answer. What does it mean to solve a system of equations? One answer is along the lines of what I think you're asking -- find exact coordinates of the points which satisfy the equation. The downside of this approach is that if you're for explicit solutions, you're going to be disappointed pretty quickly since you can't, for example, solve a general fifth degree polynomial in one variable using the basic algebraic operations of addition, subtraction, multiplication, division, and root extraction. Of course, if you are willing to relax your insistence on finding exact algebraic expressions for solutions of the system of equations, you can start to look for transcendental expressions for solutions, as well as numerical approximations to solutions.
But there's another point of view that approaches "solving a system of equations" differently. Under this point of view, if you try to solve a system of $n$ equations in $m$ unknowns, the first thing you could ask for is what is the dimension of the solution space. In general, the solution space will have dimension $\geq m - n$, but equality might not hold. And if the dimension is zero, then there are only finitely many solutions, so one might ask to count the solutions. To make the theory work out better, it is advantageous to assign multiplicities to the solutions. This is the perspective of Intersection Theory, which has as its starting point Bezout's Theorem. Two good introductions to this theory are Chapter IV of Shafarevich's Basic Algebraic Geometry I and Appendix C of Hartshorne's Algebraic Geometry.
Edit: For your specific example, you can factor $x^2 + y^2 = (x + iy)(x - iy)$ if your field $K$ has a square root of $-1$; otherwise, you can't solve $x^2 + y^2 = 0$ (in projective space).