For most of calculus, the word "function'' has meant a machine that inputs a number and outputs another number. Something like $f(x)=x^2$. But some functions have more than one input, like $f(x,y) = x^2 + y^2$, while vector-valued functions like $\vec r(t) = (\cos(t), \sin(t))$ have one input and more than one output. Each kind of function gives a different way to describe a curve in the plane.

Every ordinary function with one input and one output is associated with a graph. We usually call the input variable $x$ and the output $y$, and draw the curve $y=f(x)$. If $f(x)=x^2$, this gives us a parabola. Geometric properties of the curve, like its slope or the area under it, are related to properties of the function, like its derivative and definite integral. That's how we did one-variable calculus.

However, some curves can't be written as graphs, since they don't pass the vertical line test. This simplest example is a circle. Instead, we usually describe the unit circle by an equation: $x^2+y^2=1$. This is an example of a level set. There is a function of two variables (in this case $f(x,y)=x^2+y^2$) and we look at all of the points where $f(x,y)=1$. If we look for different values of $f$ we get different curves. (E.g. $f(x,y)=4$ is a circle of radius 2 instead of 1.)

A third way to get a curve is to track the trajectory of a moving point. If we specify both $x$ and $y$ as functions of a parameter $t$ (time), then we get more than a geometric set. We get a description of how the curve is being traced. This is called a parametrized curve or a parametric curve. (The terms are interchangable.)

Every graph can be written as a parametrized curve. For instance, we can just take $x(t)=t$, $y(t)=f(t)$, moving from left to right at constant horizontal speed, and letting the graph do the rest. However, that's not the only way to write the graph as a parametrized curve. The parametrization $x(t)=2t$, $y(t)=f(2t)$ would work just as well. It would trace out the exact same set of points, only twice as fast.

The unit circle is usually parametrized as $x(t)=\cos(t)$ and $y(t)=\sin(t)$. This traces the circle counter-clockwise, starting at $(1,0)$, going around once every $2\pi$ seconds. However, we could also start at a different point, go clockwise instead of counterclockwise, or go at a different speed. You should check that all of the following parametrizations also satisfy $x^2+y^2=1$, and so follow the unit circle:

The following video describes the three kinds of functions, and the three kinds of curves: