Rational Maps
A rational map of an elliptic curve $E(K)$ to itself is an element of the elliptic curve $E(K(E))$. (Recall $K(E)$ denotes the function field of $E$.)
Let us examine what this means. Suppose $(g_x, g_y) \in E(K(E))$, so that $g_x(X,Y), g_y(X,Y)$ are rational functions that happen to satisfy the equation for $E$. Then if $P = (x, y)$ is a point on the elliptic curve $E(K)$, we see that $(g_x(x, y), g_y(x, y))$ is also a point on an elliptic curve $E(K)$. So a rational map describes how to map points of $E(K)$ to other points of $E(K)$.
Fact: A rational map is either surjective or constant
A rational map is called an endomorphism or an isogeny if it maps the point $O$ to itself. The set of all endomorphism is denoted $\mathrm{End}(E)$.
It is not hard to check that $\mathrm{End}(E)$ is a ring (where addition is point addition, and multiplication is composition).
Identity Map
This is the pair of rational functions $X$ and $Y$.
It is also the multiplicative identity of $\mathrm{End}(E)$.
Constant Map
Let $P$ be a point $(x,y)$. Then the constant map that takes every point to $P$ is given by
(both rational functions are constants). The map $C_O$ is only constant isogeny, and it is the additive identity of $\mathrm{End}(E)$.
Translation Map
Let $P$ be a point $(x,y)$. Then the translation map that adds $P$ to every point is
Here the addition refers to point addition done over $E(K(E))$. The map can be explicitly given as follows. Set $\lambda$ to the rational function $(Yy)/(Xx)$. Let
and
Then $\tau_P = (g_x, g_y)$.
Multiplication Map
The multiplicationby$m$ map $[m]$ takes a point $P$ to $mP$, and hence it can be defined as follows $[0] = C_O, [1] = id, [m] = [m1] + [1] (m \gt 1), [m] = [m] (m \lt 0)$.
Explicit descriptions of the rational functions involved are usually given in terms of division polynomials (TODO).
Frobenius Map
Suppose we are working with a curve $E$ over $\mathbb{F}_q$. The Frobenius map is given by
It is easily verified that this is indeed an isogeny.
Twists
Certain curves have useful maps often called twists. In the following section $p$ denotes a prime.

$E: y^2 = x^3 + x$, $p = 3 \mod 4$, Twist on $E(\mathbb{F}_{p^2})$: $(X, iY)$, where $i$ is a square root of $1$.

$E: y^2 = x^3 + 1$, $p = 2 \mod 3$, Twist on $E(\mathbb{F}_{p^2})$: $(\zeta X, Y)$, where $\zeta$ is a nontrivial cube root of $1$.