Enumerative geometry is a branch of algebraic geometry trying to count finite sets related to algebraic geometry, for example "how many lines are they on a generic cubic surface", losely speaking computing degree of some zero-cycle.
Intersection theory is a branch of algebraic geometry trying to understand how to intersect cycles in a compact algebraic variety $X$, losely speaking it is the study of the Chow ring of $X$.
In some sense, enumerative geometry is a subfield of intersection theory because the methods of intersection theory turn out to be very powerful for solving enumerative problems.
I think this is explained in the introduction and the first chapters of 3264 and all that.
I am definitely not an expert but here are a few references you might want to check, more advanced (I also included intersection theory books) :
- Fulton's Intersection theory : contains all the proofs of the theorem in 3264 and more (and it's $\sim$ 500 pages).
- Katz's Enumerative Geometry and String Theory : gives a quick introduction to many subjects, and explains links with physics. The first chapters are pretty basic, but the end of the book is (relatively) advanced.
- Lectures on algebraic cycles by Bloch : the references about algebraic cycles.
- An Invitation to Quantum Cohomology: Kontsevich's Formula for Rational Plane Curves by Kock and Vainsencher. I didn't read it, but definitely Kontsevich's formula is an important problem in enumerative geometry so it might be a nice source to learn more. There are also links with enumerative tropical geometry but I'm not sure it's mentioned in the book.
Finally Donaldson-Thomas theory might be a good place to go, see this mathoverflow question. Also computation of Gromov-Witten invariants is an active research area, again here is another mathoverflow question.