Let $\phi(\boldsymbol{r})$ be a scalar field, and $\boldsymbol{a} \cdot \nabla \phi$ gives the directional derivative of $\phi$ in the direction of $a$. That is,
$$\boldsymbol{a} \cdot \nabla \phi(\boldsymbol{r}) = \lim_{t\to 0} \frac{\phi(\boldsymbol{r} + \boldsymbol{a} t) - \phi(\boldsymbol{r})}{t}$$
Now let's consider $\Phi(t) = \phi(\boldsymbol{r}_0 + \boldsymbol{a}t)$ for some finite $t$. Now, let's expand this in powers of $t$. This is a one-dimensional Taylor series.
$$\Phi(t) = \Phi(0) + \Phi'(0)t + \frac{1}{2!} \Phi''(0) t^2 + \ldots$$
To substitute back in $\Phi(t) = \phi(\boldsymbol{r}_0+\boldsymbol{a}t)$, we must compute derivatives of $\Phi$ in terms of $\phi$. Again, we resort to the basic definition of the derivative.
$$\Phi'(0) = \lim_{t\to 0} \frac{\phi(\boldsymbol{r}_0+\boldsymbol{a}t) - \phi(\boldsymbol{r}_0)}{t} = \boldsymbol{a} \cdot \nabla \phi(\boldsymbol{r})\Big|_{\boldsymbol{r}=\boldsymbol{r}_0}$$
And similarly for higher derivatives. This enables us to write,
$$\phi(\boldsymbol{r}_0+\boldsymbol{a}t) = \phi(\boldsymbol{r}_0) + [\boldsymbol{a} \cdot \nabla \phi(\boldsymbol{r})] \Big|_{\boldsymbol{r}=\boldsymbol{r}_0} t + \frac{1}{2!} [\boldsymbol{a} \cdot \nabla][\boldsymbol{a} \cdot \nabla]\phi(\boldsymbol{r}) \Big|_{\boldsymbol{r}=\boldsymbol{r}_0} t^2 + \ldots$$
It is not difficult to show that this form reproduces the form of the original question. Take $t=1$ and let $\boldsymbol{a} = (x-x_0, y-y_0)$ and $\boldsymbol{r}_0 = (x_0, y_0)$. Thus, we have built multivariate Taylor series from the well-established case of a single variable, just by use of the directional derivative.
The definition of Taylor polynomial: is the only polynomial of degree $n$ that coincides with the function and their derivatives up to the $n$-th:
$$P_n(a)=f(a)$$
$$P_n'(a)=f'(a)$$
$$P_n''(a)=f''(a)$$
$$\cdots$$
$$P_n^{(n)}(a)=f^{(n)}(a)$$
Write
$$P_n(x)=c_0+c_1(x-a)+\cdots c_n(x-a)^n,$$
and impose the $(n+1)$ conditions to find the $c_k$.
Best Answer
It is a first order approximation because the polynomial used to approximate $ f(z) $ is first order (i.e. of degree 1). This is simply a name for the approximation, so when we say we want the second order approximation, we are looking for the Taylor series written to more terms.