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 $H^*$.
Remark: In the book of Eisenbud/Harris they speak about Weil divisors/Cartier divisors and invertible sheaves and also the projective bundle formula: For any rank $e$ locally trivial sheaf $E$ on a scheme $X$, they prove there is an isomorphism of rings
$$ \operatorname{CH}^*(\mathbb{P}(E^*)) \cong \operatorname{CH}^*(X)[t]/(t^{e+1})$$
They also speak of the relation between divisors and global sections of invertible sheaves and this exposition is a bit "more detailed" than the one in Hartshorne. The projective bundle is a surjective morphism
$$\pi: \mathbb{P}(E^*) \rightarrow X$$
with the property that for any point $x\in X$ it follows the fiber
$$\pi^{-1}(x)\cong \mathbb{P}^e_{\kappa(x)}$$
is projective $e$-space on the residue field $\kappa(x)$. If $X$ is a complex projective manifold and $x$ is a closed point it follows the fiber is complex projective $e$-space. Hence $\pi$ is a fibration with fibers projective spaces of the same dimension. Some authors use the projective bundle formula to define Chern classes in the Chow ring. Hence the Theorem referred to above says that these definitions give the same result. The "degeneralcy loci approach" is more geometric, the "projective bundle formula approach" is more abstract - if you need this result in your research you should of course convince yourself that the result is correct. It is a classical and much used result.
Example. If $C:=\mathbb{P}^1_k$ is the complex projective line and $L(d):=\mathcal{O}(d)$ with $d\in \mathbb{Z}$ it follows the global sections
$$H^0(C, L(d)) \cong k[x_0,x_1]_d$$
is the vector space of homogeneous polynomials $s$ in $x_0,x_1$ of degree $d$. If $d=2$ you get
$$s:=ax_0^2+bx_0x_1 +cx_1^2$$
and if you choose the open set $U_0:=D(x_0) \subseteq C$ you get
$$ s_{U_0}=(a+bt+ct^2)x_0^2:=f(t)x_0^2 =(t-u)(t-v)x_0^2$$
and to $s$ you get the divisor $[u]+[v]$ corresponding to the roots $u,v$ of the polynomial $f(t)$. More generally if $s_{U_0}:=f(t)x_0^d$ is a global section of $L(d)$ and you write
$$ s_{U_0}=\prod_i (t-u_i)^{l_i}x_0^d$$
you get the divisor
$$ \sum_i l_i[u_i]\in Cl(C)$$
in the group of Weil divisors on $C$. The book is more than 700 pages long and give all details - again you should check these details if you need this construction. In Example II.8.20 in Hartshorne they write down the following exact sequence defining the tangent bundle of projective space $X:=\mathbb{P}^n_k$:
$$ 0 \rightarrow \mathcal{O}_X \rightarrow \mathcal{O}_X(1)^{n+1} \rightarrow T_X \rightarrow 0$$
and this sequence gives some information on the global sections of $T_X$:
$$ 0 \rightarrow H^0(X, \mathcal{O}_X) \rightarrow H^0(X, \mathcal{O}_X(1)^{n+1}) \rightarrow H^0(X, T_X) \rightarrow 0$$
and $H^0(X, \mathcal{O}_X(1)) \neq 0$. Hence the above sequence give non-trival global algebraic vector fields on projective space.
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.
Note: The books referred to above are in total $496 + 633 + 813 = 1942 $ pages and you can't yourself verify all results presented in the books. The results presented are classical results on characteristic classes and most of them are correct I believe.