Let $X$ be a surface, $C,D$ be curves on $X$ and ley $p \in C\cap D$. In Hartshorne book, $C$ and $D$ intersect transversally at $p$ if the local equations $f,g$ on $C,D$ at $p$ generate the maximal ideal $\mathcal{m}_p$ of $\mathcal{O}_{X,p}$.
But I don't understand this definition... why this implies that $T_p(C) \neq T_p(D)$??
You can think of it intuitively as the maximal ideal is functions vanishing at that point. Since it's a surface (hartshorne assumes surfaces are smooth I think), there should be two dimensions of functions vanishing at the point. Each of the tangent spaces $T_p(C)$ and $T_p(D)$ should be one dimensional at least if the curves are smooth. So if the two 1-dimensional tangent spaces generate a 2-dimensional thing, then they must not be equal or else they would only be giving a one dimensional thing.