Field lines are imaginary lines showing the direction of vectors in a (mathematical) vector field (as well as any physical field the mathematical field represents), often included in illustrations of fields. In an illustration, labeling might reveal which of the two possible directions the vectors are pointed, since simple lines don't reveal that. Field lines are commonly used to illustrate magnetic fields and often the term is used to describe phenomena that follow paths that could form a field line, e.g., "along the field lines, ...". A simple experiment of a magnet under a piece of paper with (tiny stick-like) iron filings sitting on top results in the filings illustrating the field in a "field line" manner.
Field lines can also be used to illustrate other mathematical and physical fields, including electric fields and gravitational fields.