This is when a hooked fish decides to be as far away from the angler as it is capable of going. Bonefish are legendary for running at high speed.
For others 'run' is an exaggeration and the fish just plods along like a Kenworth truck in first gear. Some fish are 'clean' and just head for the horizon. Others are 'dirty' and run for the nearest rock to snag the line on. Some fish will also jump/leap during a run.