For questions regarding the plotting or graphing of functions. For questions about the kinds of graphs with vertices and edges, use the (graph-theory) tag instead.
Given a real-valued function $f\colon \mathbf{R} \to \mathbf{R}$, the graph of $f$ is the set of all input-output pairs $(x,f(x))$ regarded as a set of points in the plane $\mathbf{R} \times \mathbf{R}$. Considering the graph of a function gives us a geometric perspective on the data that the function represents.
If the function $f$ is continuous, the graph of $f$ "looks continuous." That is, there are no gaps, and the graph is a connected curve.
If the function $f$ is differentiable, then it will contain no "sharp corners."
If we're thinking of the domain of the function as representing time, the the graph gives us a nice visualization of the change in outputs of the function over time.
A graph can be defined much more generally though. Let $\mathbf{k}$ be a local field, and suppose $f$ is a vector-valued function $f\colon \mathbf{k}^n \to \mathbf{k}^m$ where $f(x_1, \dotsc, x_n) = (y_1, \dotsc, y_m)$ and each coordinate $y_i$ of the output is a function of the $x_1, \dotsc, x_n$. In this setting, the graph of $f$ is the set of points
$$(x_1, \dotsc, x_n, y_1, \dotsc, y_m) \subset \mathbf{k}^{n+m}\,.$$
This general construction of the graph of a function can be useful in the study of algebraic geometry or the study of manifolds.
