In the formula for the Reimann tensor
Wikipedia says that $$∂_μ=\frac{∂}{∂x^μ}$$ and that they are coordianates of a vector field. But does it just mean the partial derivative of what comes after is with respect to $x^μ$ or something different, if so what? Thanks.
If $V$ is a vector field on a manifold then $V$ is a derivation on smooth functions of the manifold. It turns out that locally this means $V$ is expressed as a sum of partial derivatives with respect to manifold coordinates. In particular, $$ V = \sum_{i=1}^n V(x^i) \partial_i $$ The coordinate partial derivatives provide a convenient basis to express vector fields. The Riemann tensor you mention in your question is actually just the component of the Riemann tensor. In particular, $$ R = \sum_{\rho, \alpha, \beta, \gamma} R^{\rho}_{ \ \alpha \beta \gamma} \partial_{\rho} \otimes dx^{\alpha} \otimes dx^{\beta}\otimes dx^{\gamma} $$ Then, $$ R(dx^{\rho}, \partial_{\alpha}, \partial_{\beta}, \partial_{\gamma}) = R^{\rho}_{ \ \alpha \beta \gamma} $$. In short, $\partial_1$ plays the same role as $\hat{i}$ in multivariable calculus.