Concrete and simple applications for bipartite graphs
Assignment Problem would be one such example:


  There are a number of agents and a number of tasks. Any agent can be
  assigned to perform any task, incurring some cost that may vary
  depending on the agent-task assignment. It is required to perform all
  tasks by assigning exactly one agent to each task and exactly one task
  to each agent in such a way that the total cost of the assignment is
  minimized.


Hall's Marriage Theorem would be another:


  Imagine two groups; one of n men, and one of n women. For each woman,
  there is a subset of the men, any one of which she would happily
  marry; and any man would be happy to marry a woman who wants to marry
  him. Consider whether it is possible to pair up (in marriage) the men
  and women so that every person is happy.


The Mutilated Chessboard Problem can be solved using The Hall's Theorem:


  Suppose a standard 8x8 chessboard has two diagonally opposite corners
  removed, leaving 62 squares. Is it possible to place 31 dominoes of
  size 2x1 so as to cover all of these squares?