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$ \Gamma_{ijk} \equiv -\bigl(\partial_{i}g_{jk}-\partial_{k}g_{ij}-\partial_{j}g_{ki}\bigr)$
Christoffel symbols $\Gamma^i_{jk}$ are a particular type of connection that a Lorentzian manifold has (called the Levi-Civita connection).
A connection is a tool that we use to parallel transport tangent vectors around the manifold.
Parallel transport is just the simplest way to compare vectors at different points in the manifold.
Parallel is necssecary, for example, to define the covariant derivative.