How is $\nabla_XY$ calculated along $X$?

Question

Let $X,Y:M\to TM$ be vector fields. I know the expression of the covariant derivative $\nabla_XY|_p$ as a limit given its parallel transport on a curve along the direction of $X$. $$\nabla_XY|_p=\lim_{t\to 0}\frac{\Pi_{-tX}(Y_{\phi^X(t)})-Y_p}{t}$$ the map $\Pi_{-tX}$ being a parallel transport $\Pi_{-tX}:T_{\phi^X(t)}M\to T_pM$, the curve satisfying: $$\phi^X(0)=p \; \; ; \; \; (\phi^X(0))'=X_p$$ so what curve is $\phi^X(t)$ exactly? an integral "flow" curve of $X$ passing through $p$? a geodesic in the direction of $X$? or what?