Red line was the maximum velocity that a starship's engines could operate within normal safety protocols. Though a ship could go faster than the red line, it was not recommended and could overly stress the ship's space frame.
In 2364, when pursued by Q, Lieutenant Worf informed Captain Jean-Luc Picard that the engines were operating at warp 9.3, the red line for Galaxy-class starships. The crew was able to reach warp 9.6, at which point the captain ordered a reduction in velocity. (TNG: "Encounter at Farpoint")