Field lines are useful for visualizing vector fields, which consist of a separate individual vector for every location in space. If the vector field describes a velocity field, then the field lines follow stream lines in the flow. Perhaps the most familiar example of a vector field described by field lines is the magnetic field, which is often depicted using field lines emanating from a magnet.
A complete description of the geometry of all the field lines of a vector field is sufficient to completely specify the direction of the vector field everywhere. In order to also depict the magnitude, a selection of field lines is drawn such that the density of field lines (number of field lines per unit area) at any location is proportional to the magnitude of the vector field at that point.
Field lines can be used to trace familiar quantities from vector calculus: divergence may be seen as a net geometric divergence of field lines away from (or convergence toward) a small region, and the curl may be seen as a helical shape of field lines.
While field lines are a "mere" mathematical construction, in some circumstance they take on physical significance. 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.