A warp current, or subspace current, is an area of where subspace is in a state of motion relative to normal space. The currents affect the relative velocity of starships traveling within them.
A ship traveling with the flow would have the effect of the warp field multiplied, allowing a ship to travel at a faster relative velocity. A ship traveling against a current would have its warp field weakened, requiring a stronger warp field to overcome the current's effects.
In 2377, the USS Baldwin encountered a 25,000 rel current within Cardassian space. As a current that strong would not allow the ship to drop out of warp, Captain Patrick Ingrum allowed Baldwin to ride the current until it ended at a subspace sandbar in the Beta Quadrant. In the sandbar, Baldwin found the starship Discovery. In order to escape this sandbar, a starship would have to maintain warp 9.996 to overcome the currents effects, something which Discovery was incapable of. (USS Baldwin: "Against the Wind")