A curve in the complex plane is defined as a continuous function from a closed interval of the real line to the complex plane $z:[a,b]\to\mathbb C$.
This is just a locus definition and can be mapped to the real plane. $$\int_a^bf(t)\,dt=\int_a^b(u(t)+iv(t))\,dt=\int_a^bu(t)\,dt+i\int_a^bv(t)\,dt$$ I do not understand why we need to name these sets of integrals contours integrals when it is just vector calculus and parametric integration because complex numbers are just vectors. We could just integrate over the real plane and get our answer. Why couldn't we just call it that? I am confused as to the significance of referring to these as "contours" and having a special integral sign. $$\int_\gamma f(z)\,dz=\int_a^bf(\gamma(t))\gamma'(t)\,dt$$
I will elaborate on Izaak's comment.
Note that if $f=u+iv$ and $\gamma=\gamma_1 +i\gamma_2$, you get $$ \begin{align} \int_\gamma f(z) dz &= \int_a^b f(\gamma(t))\gamma'(t)\,dt \\ &= \int_a^b (u+iv)(\gamma(t))(\gamma_1'+i\gamma_2')(t)\,dt \\ &= \int_a^b [u(\gamma(t))\gamma_1'(t)-v(\gamma(t))\gamma_2'(t)]\,dt +i\int_a^b [u(\gamma(t))\gamma_2'(t)+v(\gamma(t))\gamma_1'(t)]\,dt \end{align} $$ which is not quite the same as what you would get in vector calculus. Treating $f$ and $\gamma$ as vector functions, you wouldn't be able to write the integrand $f(\gamma(t))\gamma'(t)$ because that would require you to know how to multiply the vectors $f(\gamma(t))$ and $\gamma'(t)$. In vector calculus instead of the complex multiplication you use a dot product: $$ \begin{align} \int_\gamma f\cdot d\vec s &=\int_\gamma f(\gamma(t))\cdot \gamma'(t)\,dt \\ &=\int_a^b u(\gamma(t))\gamma_1'(t)\,dt +\int_a^b v(\gamma(t))\gamma_2'(t)\,dt \end{align} $$