"Tailwind" is a term used to describe the movement of air coming from a direction in motion with an aircraft. In other words, if an aircraft is flying in a northerly direction with a tailwind, the wind is blowing in a northern direction so that the wind hits the back of the plane first and the front of the plane last. Planes taking-off will use the runway in opposition to the tailwind to create a headwind, which is more favorable for takeoff.
A tailwind can present many problems for a plane. When a plane flies, it moves through the air at a certain speed, and the air moves past the plane and under the wings at a certain speed. At takeoff, before the plane starts to move, if there is a tailwind, the wind is moving under the wings but in a negative direction, causing the plane to, once it starts moving, feel like it is moving slower. This forces the plane to work harder. For example, if you are running on a treadmill at 5 mph, you do not move because the ground is moving in the opposite direction at 5 mph. If you speed up to 6 mph and leave the treadmill at the same speed, then you will start to move forward on the treadmill at a rate of about 1 mph. This is how a tailwind works. If the plane has a 5 knot tailwind, and it begins its roll, when the plane reaches 5 knots in speed it will act as if it’s moving at 0 knots.
Tailwinds are also present in microbursts, which are a leading cause of airplane disasters. They can force an airplane to increase their rotating speed (lift-off speed), and a tailwind can also force an aircraft coming in to land to increase their landing speed. When in actual cruise flight, it can slow the airplane's true airspeed (which is the speed of the plane through the air).