Physical Significance
While field lines are a "mere" mathematical construction, in some circumstances they take on physical significance. In fluid mechanics, the velocity field lines (streamlines) represent the paths of particles of the fluid. In the context of plasma physics, electrons or ions that happen to be on the same field line interact strongly, while particles on different field lines in general do not interact. This is the same behavior that the particles of iron filings exhibit in a magnetic field.
The iron filings in the photo appear to be aligning themselves with discrete field lines, but the situation is more complex. It is easy to visualize as a two stage-process: first, the filings are spread evenly over the magnetic field but all aligned in the direction of the field. Then, based on the scale and ferromagnetic properties of the filings they damp the field to either side, creating the apparent spaces between the lines that we see. Of course the two stages described here happen concurrently until an equilibrium is achieved. Because the intrinsic magnetism of the filings modifies the field, the lines shown by the filings are only an approximation of the equipotential lines of the original magnetic field. Magnetic fields are continuous, and do not have discrete lines.
Read more about this topic: Field Line
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