Christoffel symbols
The connection coefficients Γ^ρ_{μν} = (1/2) g^{ρσ}(∂_μ g_{νσ} + ∂_ν g_{μσ} − ∂_σ g_{μν}) computed from the metric. Not tensors — they don't transform tensorially — but appear in covariant derivatives and the geodesic equation. Define the unique torsion-free metric-compatible Levi-Civita connection.
Definition
The Christoffel symbols Γ^ρ_{μν} are the connection coefficients computed directly from the metric tensor by the formula Γ^ρ_{μν} = (1/2) g^{ρσ}(∂_μ g_{νσ} + ∂_ν g_{μσ} − ∂_σ g_{μν}). They are symmetric in their lower two indices, Γ^ρ_{μν} = Γ^ρ_{νμ}, which is the torsion-free condition; together with metric-compatibility (∇_λ g_{μν} = 0), this picks out a unique connection from the infinite family of possible affine connections on a Riemannian manifold — the Levi-Civita connection, named after Tullio Levi-Civita's 1917 introduction of the geometric notion of parallel transport on a general manifold.
Christoffel symbols are not tensors. Their transformation law under a coordinate change picks up an inhomogeneous term involving second derivatives of the coordinate transformation, which is exactly the term needed to make covariant derivatives transform tensorially. They appear in the covariant derivative of a contravariant vector, ∇_μ V^ρ = ∂_μ V^ρ + Γ^ρ_{μν} V^ν, and with a minus sign in the covariant derivative of a covariant one. Most physically, they appear in the geodesic equation d²x^μ/dλ² + Γ^μ_{αβ} (dx^α/dλ)(dx^β/dλ) = 0, where they encode the gravitational acceleration that a freely-falling particle experiences relative to a coordinate-line frame — the apparent force of gravity recast as a curvature artefact of the chosen chart.