Gravity assist
In orbital mechanics and aerospace engineering, a gravitational slingshot is the use of the motion of a planet to alter the path and speed of an interplanetary spacecraft. It is a commonly used maneuver for visiting the outer planets, which would otherwise be prohibitively expensive, if not impossible, to reach with current technologies.
Consider a spacecraft on a trajectory that will take it close to a planet, say Jupiter. As the spacecraft approaches the planet, Jupiter's gravity will pull on the spacecraft, speeding it up. After passing the planet, the gravity will continue pulling on the spacecraft, slowing it down. The net effect on the speed is zero, although the direction may have changed in the process.
So where is the slingshot? The key is to remember that the planets are not standing still, they are moving in their orbits around the Sun. Thus while the speed of the spacecraft has remained the same as measured with reference to Jupiter, the initial and final speeds may be quite different as measured in the Sun's frame of reference. Depending on the direction of the outbound leg of the trajectory, the spacecraft can gain a significant fraction of the orbital speed of the planet. In the case of Jupiter, this is over 13 km/s. A slingshot may be simulated by rolling a steel ball past a magnet in one hand that is then moved away. Because both masses must not cross paths, the acceleration is oblique to the field and thus is similar to a sail vehicle tacking to work against the force.
It is important to understand how spacecraft move from planet to planet. The simplest way to solve this problem is to use a Hohmann transfer orbit, an elliptical orbit with the Earth at perigee and Mars at apogee. If you arrange the timing correctly the spacecraft will arrive at the outer end of its orbit right as Mars is passing by. These types of transfers are commonly used, e.g., for moving between orbits over the Earth, Earth-Moon and Earth-Mars transfers.
A Hohmann transfer to the outer planets requires long times and considerable "delta V", the sum of the changes in velocity needed at either end of the transfer orbit. This is where the slingshot finds its most common applications. Instead of the Hohmann trajectory directly to, say Saturn, the spacecraft is instead sent in a path that is aimed only as far as Jupiter, and the slingshot is then used to accelerate the spacecraft on towards Saturn. In doing so, even small amounts of fuel spent in positioning and accelerating the spacecraft on its way to Jupiter will be magnified many times once it arrives. Such missions require careful timing, which is why you often hear references to a launch window when discussing them.
A Hohmann transfer to Saturn would require a total of 15.7 km/s delta V, which is not within the capabilities of our current spacecraft boosters. A trip using multiple gravitational assists may take longer, but will use considerably less delta V, allowing a much larger spacecraft to be sent. Such a strategy was used on the Cassini probe, which was sent past Venus, Venus again, Earth, and finally Jupiter on the way to Saturn. The 6.7-year transit is slightly longer than the six years needed for a Hohmann transfer, but cut the total amount of delta V needed to about 2 km/s, so much that the large and heavy Cassini was able to reach Saturn even with the small boosters available.
Another example is Ulysses, the ESA spacecraft which studied the polar regions of the sun. All the planets orbit more or less in a plane aligned with the equator of the sun: to move to an orbit passing over the poles of the sun, the spacecraft would have to change its 30 km/s of the Earth's orbit to another trajectory at right angles to the plane of the Earth's orbit, a task impossible with current spacecraft propulsion systems. Instead the craft was sent towards Jupiter, aimed to arrive at a point in space just "in front" and "below" the planet.
As it passed the planet, the probe 'fell' through Jupiter's gravity field, borrowing a minute amount of momentum from the planet; after it had passed Jupiter, the velocity change had bent the probe's trajectory up out of the plane of the planetary orbits, placing it in an orbit that passed over the poles of the sun, rendering that region visible to the probe. All this required was the amount of fuel needed to send Ulysses to a point near Jupiter, well within current technologies.