What is the word for a path that is made naturally by the action of people walking?

You probably heard one of:

1. Desire line: A path that pedestrians take informally, rather than taking a sidewalk or set route; e.g. a well-worn ribbon of dirt that one sees cutting across a patch of grass, or paths in the snow.
2. Well-trodden path: Describing a route or path that is frequently used.
3. Beaten track: A well-populated area or well-trodden path; any busy area.

For it to remain this tantalisingly elusive, I dare say that you heard option numero uno.

Besides path, pathlet, pathway, and the many things related to ways, tracks, and trails by having those words in their names (such as runway, highway, race track), we still have many possibilities:

_{(All cited definitions from the OED.)}

• ambage: Circuits, windings, circuitous paths.
• bridle-path, -road, -way: a path fit for the passage of a horse, but not of vehicles;
• by-path, bypath: A side path, as opposed to the highroad; a private, retired, or unfrequented path.
• chare: Local name for a narrow lane, alley, or wynd, in Newcastle and some neighbouring towns; also for some country lanes and field tracks, e.g. the three which converge at Chare ends, by the landing-place on Holy Island.
• estrade: In the Brazilian rubber trade, a winding path or road connecting a series of trees. Also in Fr. form estrade.
• fare: A road, track b. spec. The track of a hare or rabbit
• feute: The traces or track (of an animal).
• footing: A mark or impression left by the foot; a footprint, or footprints collectively; a trace, track, trail.
• foot-path, footpath: A path for foot-passengers only.
• footway: A way or path for foot-passengers only.
• fostal: (pl. fostalx) The track of a hare. [apparently a contraction of footstall]
• going: Means of access; a path, road; a passage, gangway (in a church).
• gutter: A furrow or track made by running water.
• hag-path: a path through a copse.
• hollow-way: a way, road, or path, through a defile or cutting.
• ladel: A little path, by-path.
• land-way: A way or path over land. A road giving access to land.
• meanders: pl. Crooked or winding paths (of a maze); labyrinthine passages; windings or convolutions (of a vein, fissure, line, etc.).
• pad: A path, track; the road, the way.
• piste: The beaten track of a horse or other animal; the track of a race-course or training-ground. Also in extended use.
• rack: A (narrow) path or track.
• rake: A way, path; esp. a rough path over a hill, a narrow path up a clef.
• ride-way: A bridle-path.
• roddin, rodding: a path: see sheep-rodding.
• rut: A (deep) furrow or track made in the ground, esp. in a soft road, by the passage of a wheeled vehicle or vehicles.
• sithe: A going, journey, path, way.
• serpentine: A winding path or line.
• slot: The track or trail of an animal, esp. a deer, as shown by the marks of the foot; sometimes misapplied to the scent of an animal; hence generally, track, trace, or trail.
• stight: a path.
• sty: A path or narrow way.
• switchback: Applied to a railway consisting of a series of steep alternate ascents and descents, on which the train or car runs partly or who lly by the force of gravity, the momentum of each descent carrying it up the succeeding ascent; esp. to such a railway constructed for amusement at a pleasure-resort. Hence transf. of a road having steep alternate ascents and descents. sb. A switchback railway (in either sense); also tran sf. and fig.; applied in N. Amer. to a tight bend on an ascending road or trail.
• terrie, terry: A trodden path, sometimes a balk or ridge of earth separating fields or allotments.
• trace: The way or path which anything takes; course, road.
• trench: A path or track cut through a wood or forest; an alley; a hollow walk. A long and narrow hollow cut out of the ground, a cutting; a ditch, fosse; a deep furrow.
• traverse: A passage by which one may traverse or cross; a way, pass; a crossing.
• twitten: A narrow path or passage between two walls or hedges.
• upgang: An ascent, an upward path or way.

And then there is the adjective:

• wilsome: Chiefly of a way or path: Leading astray as through wild and desolate regions; hence, desert, lonely and wild; dreary.

Some of those are now rare or obsolete, or restricted to certain dialects or locales.

Some paths as shown are called cowpaths. For some discussion of cowpath and calf-path see Laure's answer to a previous question. I've heard deer trail in conversation much more frequently than cowpath, but deer trails usually are less obvious than the trail shown in the photo.