Emergency impulse power was a form of emergency power for impulse engines.
In 2267, with no warp power, Lieutenant Commander Spock ordered the activation of emergency impulse power to break free from the planet killer's tractor beam and stay ahead of the pursuing doomsday machine. The USS Enterprise was fast enough to stay ahead but doing so would exhaust their fuel supplies in less than a day. (TOS: "The Doomsday Machine")
Dialogue from several episodes, including "Where No Man Has Gone Before" and "The Doomsday Machine," suggests that the impulse drive could be used in some manner to propel a ship at faster than light speeds, albeit with a lower maximum speed and a higher rate of fuel consumption than the main warp drive. This could possibly make the relationship between impulse and warp drive analogous to the relationship between electric motors and internal combustion engines in a modern hybrid car. It would also explain Scott's remark in "Balance of Terror" about the Romulans' power being "simple impulse" and allow it to be reconciled with their clear capacity for interstellar travel.