If the structure group of the theory is $G$, with Lie algebra $\mathfrak g$, then the (local) Yang-Mills fields, are locally defined differential $1$-forms that take value in the Lie algebra $\mathfrak g$.
If the local gauge neighborhood is $U$, and we suppose that it is also a coordinate neighborhood, then we may write $$ A(x)=A_\mu^a(x)\mathrm dx^\mu\otimes t_a. $$
If spacetime is $n$ dimensional, and $G$ is $N$ dimensional, then the $A^a_\mu(x)$ are $n\times N$ local functions on the gauge domain $U$ that constitute the component functions of the local Yang-Mills field $A$.
For each value of the indices $\mu$ and $a$, $A^a_\mu$ is an ordinary real-valued function. The expression $A_\mu=A^a_\mu t_a$ (for a fixed value of $\mu$) is a $\mathfrak g$-valued local function. Usually in physics, $G$ is a matrix group, and $\mathfrak g$ is a matrix Lie algebra, in this case the generators $t_a$ are matrices, so $A_\mu^a t_a$ is (also) a matrix-valued function.