Let's briefly recall that, for a sequence $$$a_0, a_1, \dots$$$, it's ordinary generating function (OGF) is defined as a formal power series
$$$ F(x) = \sum\limits_k a_k x^k, $$$and its exponential generating function (EGF) is defined as
$$$ F(x) = \sum\limits_k a_k \frac{x^k}{k!}. $$$Generating functions are used because it's very simple to represent convolutions with them.
For two sequences $$$a_0, a_1, \dots$$$ and $$$b_0, b_1, \dots$$$, their convolution is defined as a sequence $$$c_0, c_1, \dots$$$ such that
$$$ c_k = \sum\limits_{i+j=k} a_i b_j. $$$Conversely, their binomial convolution is defined as a sequence $$$c_0, c_1, \dots$$$, such that
$$$ c_k = \sum\limits_{i+j=k} \binom{k}{i} a_i b_j. $$$If $$$A(x)$$$ and $$$B(x)$$$ are the OGFs of $$$a_i$$$ and $$$b_j$$$ then $$$A(x) B(x)$$$ is the OGF of their convolution. Conversely, if $$$A(x)$$$ and $$$B(x)$$$ are the EGFs of $$$a_i$$$ and $$$b_j$$$, then $$$A(x) B(x)$$$ is the EGF of their binomial convolution. You can read more about it here.
We will often need to extract the coefficient near $$$x^k$$$ in the expansion of an OGF $$$A(x)$$$. The conventional notation for this is
$$$ [x^k] A(x) = a_k. $$$Note that if $$$A(x)$$$ is an EGF, it will return $$$\frac{a_k}{k!}$$$ instead. To avoid this, for an EGF $$$A(x)$$$ we write
$$$ \left[\frac{x^k}{k!}\right] A(x) = a_k, $$$meaning that $$$a_k$$$ is the coefficient near $$$\frac{x^k}{k!}$$$ in the expansion of $$$A(x)$$$.
The concepts above naturally generalize for multivariate generating functions.
For example, we can consider a two-dimensional sequence $$$a_{ij}$$$ and its bivariate OGF $$$A(x, y)$$$ defined as
$$$ A(x, y) = \sum\limits_{i, j} a_{ij} x^i y^j. $$$Such two-dimensional sequences are often useful, for example, when analyzing sequence of polynomials. When extracting coefficients, it is possible to extract a coefficient near specific pair of powers $$$x^i y^j$$$, as well as near specific power $$$x^i$$$. In these cases,
$$$\begin{align} [x^i y^j] A(x, y) &= a_{ij}, \\ [x^i] A(x, y) &= \sum\limits_j a_{ij} y^j. \end{align}$$$In other words, $$$[x^i y^j]$$$ would extract the specific coefficient, while $$$[x^i]$$$ would treat $$$A(x, y)$$$ as a power series over $$$x$$$, whose coefficients are power series over $$$y$$$, and extract the coefficient near $$$x^i$$$, which in itself is a power series over $$$y$$$.