In my lecture notes I have two definitions of the Christoffel symbols. The first is the smooth functions $\Gamma^k_{ij}: U\subseteq M\to\mathbb{R}$ defined for $i,j,k=1,2$ by
$\Gamma^k_{ij}=\dfrac{1}{2}\Sigma_{l=1,2}(g^{-1})^{lk}(\dfrac{\partial g_{jl}}{\partial u_i}+\dfrac{\partial g_{li}}{\partial u_j}-\dfrac{\partial g_{ij}}{\partial u_l})$
The second as $\nabla_{\frac{\partial}{\partial u_i}}(\dfrac{\partial}{\partial u_j})=\Sigma_k \Gamma^k_{ij}\dfrac{\partial}{\partial u_k}$ where $\nabla$ is the Levi-Civita connection.
I see that the first one is a good way of computing the Christoffel symbols, but I have no idea what it means and how it relates to the second definition. Could anybody try to explain how they are related and what exactly these symbols are for?