Let $X$ be a Riemann surface. According to Griffiths and Harris "Principles of Algebraic Geometry", the complexified tangent space of $X$ at $p\in X$ is \begin{equation} \begin{split} T_{\mathbb{C},p}X=\{\xi:\mathcal{C}^{\infty}(X,\mathbb{C})_p\rightarrow\mathbb{C}:\xi(\lambda f+\mu g)=\lambda \xi(f)+\mu\xi(g)\text{ and}\\ \xi(fg)=g(p)\xi(f)+f(p)\xi(g)\text{ }\forall f,g\in\mathcal{C}^\infty(X,\mathbb{C})_p,\text{ }\forall\lambda,\mu\in\mathbb{C}\}. \end{split} \end{equation} We can consider its dual $T^*_{\mathbb{C},p}X=\{\alpha:T_{\mathbb{C},p}X\rightarrow \mathbb{C}:\alpha\text{ is $\mathbb{C}$-linear}\}$. Then, $\mathcal{C}^\infty$ $1$-forms are the sections of the bundle, $$ T^*_\mathbb{C}X=\{(p,\alpha):p\in X, \alpha\in T^*_{\mathbb{C},p}X\}. $$ In a chart they have the form $$ f\cdot\mathrm{d}x+g\cdot\mathrm{d}y $$ for some complex-valued $\mathcal{C}^\infty$ functions $f,g$.
On the other hand, we have the subbundle $$ \{(p,\alpha):p\in X, \alpha\in\mathbb{C}\cdot (\mathrm{d}z)_p\subseteq T^*_{\mathbb{C},p}X\}, $$ where $(\mathrm{d}z)_p=(\mathrm{d}x)_p+i\cdot(\mathrm{d}y)_p$. If I am not wrong, type $(1,0)$ $\mathcal{C}^\infty$ $1$-forms are the sections of this subbundle. Hence, in a chart they have the form $$ f\cdot dz $$ for some complex-valued $\mathcal{C}^\infty$ function $f$.
Now, holomorphic $1$-forms are of the type $$ f\cdot dz $$ for some holomorphic function $f$.
I guess they are sections of some bundle (related with the previous one). What bundle are we talking about?