Before I explain a V1 cut....I feel I should explain V speeds.
V speeds are specific defined speeds for certain segments of flight.
The first speed that comes up on takeoff is V1 followed by either V2 or VR depending on the plane.
V1 is the speed at which the takeoff should not be aborted. Aborting after V1 will likely be more dangerous (even with a shredded tire for example or an engine on fire) than continuing the takeoff and returning for landing.
Losing an engine on most multi-engine aircraft will cause the plane yaw and have a loss of performance. Losing an engine at V1 means the pilot has to quickly identify the engine that has failed and correct for it. The plane will likely yaw into the dead engine. The yaw is corrected by applying opposite rudder force. So if the left engine failed the plane would yaw left. Right rudder would be applied with the right foot. The term "dead foot, dead engine" helps. Left foot isn't doing anything...that's the side of the dead engine.
Planes with wing mounted engines tend to yaw more than fuselage mounted engines due to the distance between the engines. There's a better term for that...but right now it escapes me.
Now let's prepare for takeoff.
When I brief my takeoff I include the following statement for MOST, but not all, runways.
"We will abort for any reason up to 80 knots. After 80 knots we will only abort for engine failure, fire, safety of flight item or if the plane won't fly. After V1 we will take the plane into the air and come back for landing. I will fly the plane, you run the checklist. "
Some shorter runways or runways with rain/snow/ice can cause aborts only to 80 knots and no aborts afterwards due to the inability to stop on the remaining pavement after 80 knots.
Assume the following speeds for my example:
V1 - 125
VR - 127
V2 - 132
VR is the speed at which rotation should begin....it's not the speed at which the wheels should instantly lift off the ground.
V2 is the speed which should be flown after takeoff in the event on an engine failure.
I advance the thrust levers to takeoff power.
The 80 knot call is made and all is well.
At 125 knots the Captain calls, "V1"
Simulaneously the plane begins a yaw to the left.
I begin correcting immediately.
The Captain calls , "VR - Rotate" while I am correcting.
I use the rudder to keep the plane's nose on centerline. Done properly is takes just a few moments...seconds....it's done precisely and quickly.
A quick glance at my speed tape shows 130 knots.
I smoothly rotate the nose into the air, keeping the rudder pressure, accelerate to V2 and fly the profile.
Profiles vary signifianctly between aircraft.
Really powerful planes can climb non-stop up to a safe altitude, typically 1500 feel above field level, and run the checklist. Other planes require and intermediate level off to accelerate before climbing above 1500 feet AFL to run checklist. In addition to the level off, terrain avoidance and specific airport engine out procedures may exist.
That's the short and skinny. The V1 cut can be very difficult if the pilot doesn't smoothly and correctly fly the plane. If the pilot tries to rotate before the plane is under control it will be MUCH more difficult to fly once airborne. While still rolling down the runway the tires on the ground help stabilize the plane. If the yaw is corrected properly a fairly stable plane is brought into the air.
If the yaw isn't corrected and the plane is taken into the air, the moment the tires are off the ground the instability will only get worse. Things can get hairy fast as an unstable plane is just feet above the ground with less than optimal performance.
Hope that helps.