For the purposes of elliptic curves and pairings with affine coordinates, functions are rational functions (ratios of two polynomials) in the two variables $X$ and $Y$ with coefficients in compatible fields. Curves are the set of points where a particular function is zero. Lines are curves where the underlying function is a polynomial of total degree 1. A function on a curve (usually the curve is defined by another function) is the set of values that the function takes on points of the curve i.e. the value of the function at places where the other function is zero.
For example if we work over the rational numbers and consider the function $C(X,Y)=Y^2-X^3+X-1$. This defines the elliptic curve $E:C(X,Y)=0$ which we might write as $E:Y^2=X^3-X+1$. Consider also the function $L(X,Y)=2X-Y-1$, this defines the line $\ell:L(X,Y)=0$ which we might usually write $\ell:Y=2X-1$. Although the functions are defined for all rational values of $X$ and $Y$, we can specialise to values that lie on curves. The function $C$ on the curve $E$ is zero everywhere, but the function $L$ takes more interesting values. Consider $L$ evaluated at the point $(5,-11)$ which lies on $E$. This is 20. Likewise we can talk about the function $C$ on the "curve" $\ell$ e.g. if we take the point $(7,13)$ which lies on $\ell$ we see $C(7,13)=-168$.
Clearly we can talk about many different functions defined on $E$ and not just $L$.
There are interesting relationships between a function $f$ on a curve defined by a function $g$ and the function $g$ on a curve defined by the function $f$. These start with the observation that the zeroes are shared. In particular the zeroes of $L$ on $E$ are $(0,-1)$, $(1,1)$, and $(3,5)$ which are also the zeroes of $C$ on $\ell$.
