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Gravity assist, aka gravitational slingshot or swing-by is the use of an interaction with an astronomical body to boost the speed of a spacecraft. Gravity assists can be very effective, vastly reducing the amount of fuel needed for a space mission, in turn, reducing the rocket power needed to begin an interplanetary space flight, and they are now commonly used and make otherwise impractical missions possible. Compared with alternatives, they can require more travel time, though in some instances can shorten it. The earliest space flights outward from Earth simply used the fuel necessary to shift to the outward trajectory of a more eccentric orbit, then when the spacecraft approaches the target orbit, to shift into it, the amount of fuel necessary to do these requiring a larger rocket to launch off Earth, in turn, requiring yet more additional fuel.
Gravity assists are also used to reduce the speed of a spacecraft, and have been useful slowing down spacecraft traveling from Earth closer to the Sun, for example, to match the speed of Mercury's orbit. Gravity assist can produce extremely fuel-efficient travel throughout the inner solar system, certainly out into the Kuiper Belt, but assists can require particular launch windows, and the highest efficiencies can require a lot of travel time.
A spacecraft undergoing a gravitational interaction with a body that is moving in some direction invariably picks up some velocity along that direction, and that is what an assist uses. For example, if the flight of a spacecraft leaving Earth orbit (entering some eccentric orbit of the Sun) is arranged to interact with Venus so as to increase its orbital speed, it can enter a more eccentric orbit that carries it to a further distance from the Sun. This latter eccentric orbit (like those of comets) can travel outward among outer planets, yet then returns inward such that the same planet can be used for more gravity assists, each producing a more-eccentric orbit. Current missions often use multiple such assists, using the same and/or additional planets.
The gravitational interaction is (somewhat) analogous to bouncing a ball off a vertical surface. If the surface is simultaneously moving toward you, the ball can return after the bounce at an increased speed. (Such a moving vertical surface could be a paddle, such as in a paddle ball game: quite obviously additional speed can be imparted on the ball.) Such as bounce is like such an interaction between the spacecraft and the planet, which consists of the spacecraft approaching the planet nearly head-on, being drawn around behind it and sent back somewhat near the opposit direction (tracing a hyperbola). Using the spacecraft's booster during the interaction can adjust the resulting velocity, both to further increase the speed-change and to place the spacecraft on the desired trajectory (i.e., putting in the desired subsequent orbit).