An action
of a group $G$ on a set $X$ is transitive if it has a single orbit, i.e., (i) $X$ is inhabited and (ii) for any two elements, $x, y$, there exists $g\in G$ such that $y = g * x$.
This definition rules out the action of $G$ on the empty set. Note that transitivity of an action is sometimes defined via (ii) alone.
For $k\ge 0$, an action $G \times X \to X$ is said to be $k$-transitive if the componentwise-action $G \times X^{\underline{k}} \to X^{\underline{k}}$ is transitive, where $X^{\underline{k}}$ denotes the set of tuples of $k$ distinct points (i.e., injective functions from $\{1,\dots,k\}$ to $X$). For instance, an action of $G$ on $X$ is 3-transitive if any pair of triples $(a_1,a_2,a_3)$ and $(b_1,b_2,b_3)$ of points in $X$, where $a_i \ne a_j$ and $b_i \ne b_j$ for $i\ne j$, there exists $g \in G$ such that $(b_1,b_2,b_3) = (g a_1,g a_2,g a_3)$.
A transitive action that is also free is called regular.
A set equipped with a transitive action of $G$ (and which is inhabited) is the same thing as a connected object in the category of G-sets. A $G$-set may be decomposed uniquely as a coproduct of transitive $G$-sets.
Any group $G$ acts transitively on itself by multiplication $\cdot : G \times G \to G$, which is called the (left) regular representation of $G$.
The alternating group $A_n$ acts transitively on $\{1,\dots,n\}$ for $n \gt 2$, and in fact it acts $(n-2)$-transitively for all $n \ge 2$.
The modular group $PSL(2,\mathbb{Z})$ acts transitively on the rational projective line $\mathbb{P}^1(\mathbb{Q}) = \mathbb{Q} \cup \{\infty\}$. The projective general linear group $PGL(2,\mathbb{C})$ acts 3-transitively on the Riemann sphere $\mathbb{P}^1(\mathbb{C})$.
An action of $\mathbb{Z}$ (viewed as the free group on one generator) on a set $X$ corresponds to an arbitrary permutation $\pi : X \to X$, but the action is transitive just in case $\pi$ is a cyclic permutation.
Let $* : G \times X \to X$ be a transitive action and suppose that $X$ is inhabited. Then $*$ is equivalent to the action of $G$ by multiplication on a coset space $G/H$, where the subgroup $H$ is taken as the stabilizer subgroup
of some arbitrary element $x \in X$. In particular, the transitivity of $*$ guarantees that the $G$-equivariant map $G/H \to X$ defined by $g H \mapsto g * x$ is a bijection. (Note that although the subgroup $H = G_x$ depends on the choice of $x$, it is determined up to conjugacy, and so the coset space $G/H$ is independent of the choice of element.)
Helmut Wielandt. Finite Permutation Groups. Academic Press, 1964.
Fundamental theorem of group actions on the Group Properties Wiki.