Comment: "I'm a physicist who is trying to make sense of the relationship between the number of zeros of a section from an associated vector bundle and the Euler characteristic."
Reply: There is a theorem (Theorem 5.3) In Harris/Eisenbud's book on algebraic geometry ("3264 and all that...") which relate the chern class $c_i(E)$ of a finite rank locally free sheaf $E$ to the "degeneracy loci" $[D(\tau_i)]$ of a set of global section of $E$:
Theorem: If $E$ is a locally free sheaf on $X$ of rank $r$ and if $\tau_0,..,\tau_{r-i}\in H^0(X,E)$ are global sections with $D(\tau_i)$ the degeneracy locus where they are independent, then there is an equality
$$c_i(E)=[D(\tau_i)]\in \operatorname{CH}^i(X).$$
Here $\operatorname{CH}^i(X)$ is the i'th Chow group of $X$. The theorem is valid in general and should apply to an "algebraic variety" over the real or complex numbers. Hence if your manifold is algebraic, this result gives a relation between characteristic classes and degeneracy loci. I believe characteristic classes in various cohomology theories were initially defined using this type of method - it is a more "geometric approach" to this construction.
You may for any non-singular variety $X$ of dimension $n$ define the Euler characteristic of $X$ as follows:
$$ \chi(X):= c(T_X) \cap [C] = \int_X c_n(T_X)$$
where $T_X$ is the tangent bundle. Choose a global section $\tau_0$ of $T_X$ (you view a global section of $T_X$ as a global algebraic vector field on $X$). You get an equality
$$c_n(T_X):=[D(\tau_0)]\in \operatorname{CH}^n(X)$$
and this relates the Euler characteristic $\chi(X)$ to the degeneracy loci $D(\tau_0)$ of a vector field $\tau_0$.
There is a "cycle map"
$$ \gamma: \operatorname{CH}^*(X) \rightarrow H^*(X, \mathbb{Z})$$
where $H^*$ denotes singular cohomology, and you get an equality
$$ c_i^H(E):=\gamma(c_i(E))=\gamma([D(\tau_i)])\in H^i(X, \mathbb{Z}).$$
Hence you get a similar equality in $H^*$.
References: For algebraic varieties/schemes the relationship between Weil divisors/Cartier divisors, rational functions and invertible sheaves is explained in Hartshorne, Section II.6. For complex manifolds you may consult Griffiths/Harris "Principles of algebraic geometry", page 413 on the Gauss-Bonnet formulas II and III where the result (Theorem 5.3 above) is done for complex manifolds. They prove that the ith Chern class $c_i(E)$ of a holomorphic vector bundle $E$ on a complex manifold $M$ is Poincare dual to a "degeneracy cycle" $D_{k-r+1}$. They also prove the Gauss-Bonnet III formula that $c_n(M)=\chi(M)$. The Griffiths/Harris book is done in the language of compex manifolds and local coordinates and is easier to read for a person with background from differentiable/complex manifolds. The Chern classes and degeneracy cycles live in singular homology and cohomology groups of $M$. The Hartshorne approach is more abstract and algebraic.