An impulse engine consists of one or more nuclear fusion reactors connected to exhaust vents that generate thrust to accelerate the ship. The thrust generated by an impulse engine is typically tiny compared to the actual mass of the ship, but the use of "mass lightening" technology (involving subspace fields) reduces the effective mass of the ship enough for the thrust to provide considerable acceleration.
Because impulse engines include fusion reactors, they are typically the ship's secondary power source, outpowered only by the warp core.
Impulse engines are capable of powering a ship for short journies at FTL speeds (assuming the ship has a functioning warp drive), but doing so drains fuel rapidly. The Enterprise nearly exhausted the fuel for its impulse engines in just 14 hours of interstellar travel when the warp core was inoperable due to "burning out" the ship's dilithium crystals.
Trekkie PicardAlpha's estimate of acceleration from Federation impulse engines.