14128, It depends on when you encounter the failure. Given the journalistic ignorance of regarding the takeoff roll as "taxiing" (I kid you not...), and ergo "takeoff" begins only when the nosewheel lifts off... I'll spell it out for you:
During revenue takeoffs, all engines must be operating normally when the aircraft takes the runway. If an engine failure occurs before some predetermined speed (called V1), then the crew slam on the brakes, switch on reverse thrust, handluggage flies about the cabin, passengers scream, and the 11pm news reports on how disaster was narrowly averted. But no harm done, usually.
On the other hand, if V1 has been reached before the engine quits, then a) there is no longer enough runway remaining to stop, and b) the airplane will fly away quite nicely, because the crew have this contingency well covered in their flight planning.
Whether the flight continues to destination with a failed engine is dependent on many many variables. If it's a twin, they'll probably land pretty soon unless so much fuel is on board that they're overweight for landing.