Groupoid actions on spaces
Perhaps the most natural example is given by universal covers?
Let $X$ be a "nice" space. For a point $x\in X$ let $\tilde X_x$ be the universal covering of $X$ taken at $x$ (the fiber at $y \in X$ is the homotopy classes of paths $[0,1]\to X$ which start at $x$ and end at $y$, where we are taking homotopy classes relative to $\lbrace0,1\rbrace$).
Let $\pi$ be the fundamental groupoid of $X$. Then there is a functor $\pi\to \text{Top}$ given on objects by $x\mapsto \tilde X_x$. On morphisms of $\pi$ from $x$ to $y$, the functor is given by the map $\tilde X_x \to \tilde X_y$ that is induced by concatenating paths.
There is a nice overview of Moerdijk and Mrcun in the proceedings of the PQR 2003 conference on groupoids and their actions and stuff. So this might provide quite a number of examples from (differential) geometry. They mainly investigate Lie groupoids.
In the setting of Lie groupoids, there are also more refined notions of "actions" and many examples. Maybe you take a look there.
Personally, I have a nice (simple?) example of a groupoid action: take a bunch of algebras (associative) over a common field (or ring...) and consider the "isomorphism groupoid": the objects are the algebras, the arrows the isomorphisms between them. Then you have an obvious "action".
Slightly more interesting is the "Picard groupoid" which also acts on the algebras. Now the arrows are (iso-classes) of Morita equivalences. You can act with these two groupoids on all kind of stuff like the K-theories or the lattices of ideas of the algebras and so on...
Actions of a Lie groupoid are defined on p. 34 of K.C.H. Mackenzie, "General Theory of Lie Groupoids and Lie Algebroids" LMS Lecture Notes Series no 213, 2005.
Note that in general for a groupoid action of $G$ on sets it is often convenient to follow C. Ehresmann and to insist on having a function $f: E \to Ob(G)$ so that $g \in G(x,y)$ maps the fibre of $f$ over $x$ to the fibre over $y$. In fact a standard equivalence is between actions on sets in this sense; functors $G \to Sets$; and covering morphisms of the groupoid $G$. But the point of the first definition is that this easily transcribes to the case $E$ is a topological space, as in Mackenzie's book. A full exposition of covering space theory based on covering morphisms of groupoids, rather than actions, is given in the book now called "Topology and Groupoids", and was in the 1968 edition.
John Klein is also right to emphasise the covering space example. This leads to the idea that for the cellular homology of the universal cover of a CW-complex you actually need chain complexes with a groupoid of operators, rather than the usual group of operators. This idea was developed in a paper with Higgins (Proc Camb. Phil. Soc. (1990)) and is explained in the book Nonabelian algebraic topology, see for example Section 8.4.
Edit May 19: A simple and basic example of a groupoid acting on spaces generalises the case of a topological group acting on itself by left multiplication. A groupoid $G$ acts on the families of stars $St_G(x), x \in Ob(G)$, where $St_G(x)$ is the union of the sets $G(x,y)$ for all $y \in Ob(G)$, using the convention that if $g: z \to x, h:x \to y$ then $gh: z \to y$. If $G$ is a topological groupoid, then we get an action of $G$ on topological spaces.
An example of this is the case the space $X$ admits a universal cover. Then the fundamental groupoid $\pi_1 X$ may be topologised making it a topological groupoid. (R. Brown and G. Danesh-Naruie, ``The fundamental groupoid as a topological groupoid'', Proc. Edinburgh Math. Soc. 19 (1975) 237-244.) The star of $\pi_1 X$ at $x$ is of course the universal cover of $X$ based at $x$. (See also Topology and Groupoids 10.5.8, which deals with the case $(\pi_1 X)/N$ for $N$ a totally disconnected normal subgroupoid of $\pi_1 X$. )