The following is definition 12.2.6 in Vakil's notes.
A $k$-scheme is $k$-smooth of dimension $d$, or smooth of dimension
$d$ over $k$, if it is pure dimension $d$, and there exists a cover by
affine open sets $\operatorname{Spec} k[x_1,\dots , x_n]/(f_1,\dots, f_r)$ where the Jacobian matrix has corank $d$ at all points.
The Jacobian matrix at a point $p$ is the usual matrix of partials evaluated at $p$.
At closed points, the quotient ring is a field, so we have a map of vector spaces and the usual definition of rank applies. For a general prime $P$, what is meant here? For Vakil, a ring element $f\in R$ has the value $f\in R/P$ at the prime $P$. But in this general case we have only a map of domains, not fields, and its not clear to me what is meant by corank here.
One way to interpret this is that certain minors vanish, but I'm not sure if that was what Vakil intended. He explicitly defines "corank" as dimension of the cokernel.
Best Answer
The Jacobian criterion does not work on non-closed $\mathbb{K}$-points of a $\mathbb{K}$-scheme locally of finite type; or it does not work as we know it!
Let $X$ be a scheme over a field $\mathbb{K}$, locally of finite type; that is: \begin{gather} \forall P\in X,\,\exists U\subseteq X\,\text{open,}\,t_1,\dots,t_n\,\text{indeterminates,}\,f_1,\dots,f_r\in\mathbb{K}[t_1,\dots,t_n]=R,\\(f_1,\dots,f_r)=I: P\in U\simeq\operatorname{Spec}R_{\displaystyle/I}\cong V(I)\subseteq\mathbb{A}^n_{\mathbb{K}}\,\text{affine closed subscheme} \end{gather} therefore \begin{equation} \forall P\in U\subseteq X,\,T_PX\cong T_PU\cong T_PV(I)\leq T_P\mathbb{A}^n_{\mathbb{K}}=T_{\mathfrak{m}_P}\mathcal{O}_{\mathbb{A}^n_{\mathbb{K}},P}=\left(\mathfrak{m}_{P\displaystyle/\mathfrak{m}_P^2}\right)^{\vee}\cong\kappa(P)^n \end{equation} where $\mathfrak{m}_P$ is the maximal ideal of the local ring $\mathcal{O}_{\mathbb{A}^n_{\mathbb{K}},P}$ and $\kappa(P)$ is the relavant residue field.
If $P$ is a closed point of $X$, then $P$ is a closed point in $U$; let $\mathfrak{p}$ be the maximal ideal of $R$ corresponding to $P$, and let $\varphi:\mathbb{K}[s_1,\dots,s_r]\to\mathbb{K}[t_1,\dots,t_n]$ the morphism of $\mathbb{K}$-algebras such that \begin{equation} \forall i\in\{1,\dots,r\},\,\varphi(s_i)=f_i; \end{equation} then: \begin{gather*} \operatorname{coker}\varphi=R_{\displaystyle/I},\\ \varphi^{*}:\mathbb{A}^n_{\mathbb{K}}\to\mathbb{A}^r_{\mathbb{K}},\\ \ker\varphi^{*}=V(I); \end{gather*} let $\varphi_0=\varphi_{\varphi^{*}(P)}:\mathbb{K}[s_1,\dots,_r]_{(s_1,\dots,s_r)}\to\mathbb{K}[t_1,\dots,t_n]_{\mathfrak{p}}$, by definition: \begin{equation} (d_P\varphi_0)^{\vee}:\left(T_O\mathbb{A}^r_{\mathbb{K}}\right)^{\vee}\to \left(T_P\mathbb{A}^n_{\mathbb{K}}\right)^{\vee} \end{equation} and in particular: \begin{equation} \left(T_PV(I)\right)^{\vee}=\operatorname{coker}(d_P\varphi_0)^{\vee}\Rightarrow T_PV(I)=\ker d_P\varphi_0; \end{equation} where $(d_P\varphi_0)^{\vee}$ is a $\mathbb{K}$-linear map and $T_P\mathbb{A}^n_{\mathbb{K}}\cong\kappa(P)^n\cong\mathbb{K}^m$ by Hilbert's (Strong) Nullstellensatz (see also the REMARK1).
By Hilbert's Basis Theorem, $\mathfrak{p}$ is a finite generated $\mathbb{K}$-vector space, therefore: \begin{gather} \mathfrak{p}=(e_1,\dots,e_m)\\ \forall i\in\{1,\dots,r\},\,(d_P\varphi_0)^{\vee}(\overline{s_i})=\overline{f_i}=\sum_{j=1}^ma_i^j\overline{e_j},\,\text{where:}\,a_i^j\in\mathbb{K}; \end{gather} but every $a_i^j$ no makes sense as the formal derivation of $f_i$ with respect to the element $e_j$ computed at $P$, unless $P$ is a closed $\mathbb{K}$-point of $X$! (Jump to UPDATE.)
Again: is it all clear?
I repeat, the definition 12.2.6 and it is completed by exercise 12.2.H, it is true that this definition is intricated and in apparence is unsatisfactory; but this definition is completely right and it works on $\mathbb{K}$-schemes locally of finite type.
REMARK1: If $P$ is not a closed point of $X$, that is a closed point of $\mathbb{A}^n_{\mathbb{K}}$, we can't apply the Hilbert (Strong) Nullstellensatz.
UPDATE: By a base change, we can define $\overline{\varphi}:\mathbb{F}[s_1,\dots,s_r]\to\mathbb{F}[t_1,\dots,t_n]$ where $\mathbb{F}=\kappa(P)$, we can repeat the same reasoning described for $\varphi$ and we can prove that: \begin{equation} T_P(V(I)/\operatorname{Spec}\mathbb{F})=\ker d_P\overline{\varphi}_0 \end{equation} where the notation is clear.
Because $P$ is a closed $\mathbb{F}$-point of $\mathbb{A}^n_{\mathbb{F}}$, then: \begin{equation} \exists\alpha_1,\dots,\alpha_n\in\mathbb{F}\mid\mathfrak{p}=(t_1-\alpha_1,\dots,t_n-\alpha_n) \end{equation} and therefore, via a formal Taylor series of the $f_i$'s \begin{gather} \forall i\in\{1,\dots,r\},\,(d_P\varphi_0)^{\vee}(\overline{s_i})=\overline{f_i}=\dots=\sum_{j=1}^n\frac{\partial f_i}{\partial t_j}\bigg|_{(t_1-\alpha_1,\dots,t_n-\alpha_n)}\left(\overline{t_j-\alpha_j}\right) \end{gather} that is $T_P(V(I)/\operatorname{Spec}\mathbb{F})$ is the kernel of the linear map from $\mathbb{F}^r$ to $\mathbb{F}^n$ described from the Jacobian matrix of the $f_i$'s with respect to $t_j$'s, with entries in $\mathbb{F}$ and valued in $P$.
REMARK2: In general $T_PV(I)$ and $T_P(V(I)/\operatorname{Spec}\mathbb{F})$ are not isomorphic as $\mathbb{F}$-vector spaces; see exercise 6.3 from Görtz and Wedhorn - Algebraic Geometry I, Schemes With Examples and Exercises.
EDIT: The enumeration is refered to December 29 2015 FOAG version.