Chip-firing (aka the sandpile model, though both notions have a plethory of different meanings) and rotor-routing. Some sources (no claim of completeness):
Alexander E. Holroyd, Lionel Levine, Karola Meszaros, Yuval Peres, James Propp, David B. Wilson, Chip-Firing and Rotor-Routing on Directed Graphs, arXiv:0801.3306. If you only have time for one source, then this should probably be it; it contains most of the "fun results", including the fact that chip-firing stabilizes (when there is a sink) and the final result does not depend on the path taken.
Anders Björner, László Lovász, Chip-firing games on directed graphs and Anders Björner, László Lovász, P. W. Shor, Chip-firing games on graphs. These older works have remarkably little intersection with the previous one; they focus more on the use of greedoids and the relations to spectral graph theory.
Benjamin Bond, Lionel Levine, Abelian networks, trilogy: part I (arXiv:1309.3445), part II (arXiv:1409.0169), part III (arXiv:1409.0170). This is a vast generalization, which is probably too abstract and notation-heavy for undergrads, but it really seems to bring out the real ideas behind the material.
Bálint Hujter, Lilla Tóthmérész, Chip-firing based methods in the Riemann--Roch theory of directed graphs, arXiv:1511.03568. A new approach to the "discrete Riemann-Roch theory" of sandpiles on graphs. See also the references therein, as they occasionally give more elementary proofs.
Tian-Yi Jiang, Ziv Scully, Yan X Zhang, Motors and Impossible Firing Patterns in the Parallel Chip-Firing Game, arXiv:1211.6786. This is about a deterministic variation of the chip-firing game: All vertices have to fire as soon as they can, not just when they decide to. Corollary 7.1 is a beautiful result that, to my knowledge, has not received the nice proof it deserves (the proof given in the paper involves checking that a weighted graph with 64 vertices and 256 edges has no negative-weight cycles).
Eric Goles, Michel Morvan, Ha Duong Phan, Sandpiles and order structure of integer partitions. This relates chip-firing to the dominance order on partitions (don't ask me how).
... and many more.