= Solution
A <covariant derivative> on the <tangent bundle> is an <affine connection>: it assigns a <vector field> $\nabla_XY$ to <vector fields> $X,Y$, is linear over <smooth functions> in $X$, is real-linear in $Y$, and obeys the <Leibniz rule> $\nabla_X(aY)=X(a)Y+a\nabla_XY$. Thus it differentiates a <vector field> while taking account of how its local <basis> changes. It extends to <tensor fields> by the <Leibniz rule>, compatibility with <tensor contraction>, and $\nabla_Xa=X(a)$ for a <scalar>.
In a local <basis> $e_\mu$, the <connection components> are defined by $\nabla_{e_\rho}e_\nu=\Gamma^\mu{}_{\nu\rho}e_\mu$. In a <coordinate basis>, $e_\rho=\partial_\rho$, so
$$
(\nabla_\rho V)^\mu=\partial_\rho V^\mu+\Gamma^\mu{}_{\nu\rho}V^\nu,\qquad (\nabla_\rho\alpha)_\mu=\partial_\rho\alpha_\mu-\Gamma^\nu{}_{\mu\rho}\alpha_\nu.
$$
A <covariant derivative> of a <tensor field> has tensorial transformation properties, but the <connection components> themselves do not: changing the <coordinate basis> introduces second derivatives of the coordinate change. \b[They describe the connection in a chosen basis, rather than components of a tensor.]
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