Reid's Undergraduate Algebraic Geometry requires very very little commutative algebra; if I remember correctly, what it assumes is so basic that it is more or less what Eisenbud assumes in his Commutative Algebra!
The trouble with algebraic geometry is that it is, in its modern form, essentially just generalised commutative algebra. Indeed, in a very precise sense, a scheme can be thought of as a generalised local ring. (The structure sheaf $\mathscr{O}_X$ of a scheme $X$ is a local ring object in the sheaf topos $\textrm{Sh}(X)$, and a $\mathscr{O}_X$-module is literally a module over $\mathscr{O}_X$ in the topos.) If you are willing to restrict yourself to smooth complex varieties then it is possible to use mainly complex-analytic methods, but otherwise there has to be some input from commutative algebra.
That said, it is not necessary to learn all of Eisenbud's Commutative Algebra before starting algebraic geometry. Classical algebraic geometry, in the sense of the study of quasi-projective (irreducible) varieties over an algebraically closed field, can be studied without too much background in commutative algebra (especially if you are willing to ignore dimension theory). Reid's Undergraduate Algebraic Geometry, Chapter I of Hartshorne's Algebraic Geometry and Volume I of Shafarevich's Basic Algebraic Geometry all cover material of this kind.
Modern algebraic geometry begins with the study of schemes, and there it is important to have a thorough understanding of localisation, local rings, and modules over them. A scheme is a space which is locally isomorphic to an affine scheme, and an affine scheme is essentially the same thing as a commutative ring. The theory of affine schemes is already very rich – hence the 800 pages in Eisenbud's Commutative Algebra! For general scheme theory, the standard reference is Chapter II of Hartshorne's Algebraic Geometry, but Vakil's online notes are probably much more readable. Volume II of Shafarevich's Basic Algebraic Geometry also discusses some scheme theory. Also worth mentioning is Eisenbud and Harris's Geometry of Schemes, which is a very readable text about the geometric intuition behind the definitions of scheme theory.
Perhaps the most important piece of technology in modern algebraic geometry is sheaf cohomology. For this, some background in homological algebra is required; unfortunately, homological algebra is not quite within the scope of commutative algebra so even Eisenbud treats it very briefly. The first few chapters of Cartan and Eilenberg's Homological Algebra give a good introduction to the general theory but is strictly more than what is needed for the purposes of algebraic geometry. (For example, they allow their rings to be non-commutative.) The last chapters of Lang's Algebra also cover some homological algebra. Chapter III of Hartshorne's Algebraic Geometry is dedicated to the cohomology of coherent sheaves on (noetherian) schemes.
It is difficult to say anything not too generic but the very typical problems that appear in engineering are:
1) Can I decompose this complicated problem into easier to understand problems?
2) I understand simple relationships. Can I extrapolate these simple relationships into solutions of more complicated systems?
You've likely seen 1) as an electrical engineer. You always try to take complicated functions, such as a triangle wave, which are hard to create in nature and use your basic functions like sine and cosine to reconstruct them.
Maxwell's equations are an example of 2) but even more fundamental is Newtonian mechanics. It is easier to understand that velocity is linear than to say that a falling object has position given by a parabola. It is easier to understand that the time-change in a current in an inductor induces a proportional change in voltage than to see that the voltage is an exponential.
Back to your original question, what does this have to do with algebraic geometry? It turns out that many of the questions asked in these contexts
a) extend to the complex numbers
b) extend to projective spaces.
With appropriate adjectives inserted, analytic questions on projective spaces over the complex numbers can be phrased as algebraic questions via Serre's GAGA theorem. Writing the triangle wave as a sum of sine and cosine functions, while on the surface looks strictly analytic, actually can be deduced by working with algebraic functions on the projective complex line.
Best Answer
It's a massive subject, and there are many different perspectives; here are a few that don't require too much background.
Perspective one: It's a generalization of linear algebra.
Linear algebra is about dealing with systems of linear equations. This is easy: the set of solutions to a (homogeneous) system is just some subspace of $F^n$ (where $F$ is the field of scalars), and you can compute its dimension by row-reducing your system into echelon form.
Algebraic geometry is about dealing with systems of polynomial equations. As you may imagine, this is much harder. In linear algebra, much of the theory is entirely independent of the field $F$, at least until you want to diagonalize operators; in algebraic geometry, non-algebraically-closed $F$ are a massive headache, and there are phenomena in characteristic $p$ that don't show up in characteristic $0$.
Perspective two: It's a computational tool in classical geometry.
In geometry and topology we may wish to study invariants of manifolds. We define lots of invariants, e.g., homology groups, but how can we get our hands on them? For most examples, we can't do it easily at all, but if the example happens to be a complex manifold given by polynomial equations, there's a lot more that we can say. This is especially important if you want to do things with computers.
Perspective three: It's a conceptual way to think about commutative algebra.
If I give you some ring, OK, great, it has prime ideals, maximal ideals, zero divisors, etc. What does all this mean, and how do you ever remember the barrage of technical theorems about integrality, Artin rings, regular local rings, etc?
If the ring is the ring of functions on some space, then the geometry of the space may reflect properties of the ring, and we can remember the commutative algebra by picturing the geometry. What Grothendieck realized is that if we define "space" correctly (which is not so easy), every ring is the ring of functions on some space! For an example of how you might relate geometry to intrinsic properties of the ring: the space attached to a ring is connected if and only if all of the zero divisors in the ring are nilpotent.