The touch and go in aviation
A touch and go is a pilot's term for a practice landing, executed all the way up to touchdown with all wheels in contact with the ground and rolling, coupled with an immediate return to takeoff power settings and a rollout to liftoff. The airplane should touch the ground, roll for a moment, and go. This is useful for practicing the maximum number of approaches and landings (the most difficult part of normal flying) in the minimum amount of time, as well as without tying up the active runway for longer than necessary.
The reason for carrying on until all wheels are rolling is to ensure that the aircraft has made touchdown in proper attitude and in the appropriate speed envelope such that a rollout and stop (or turnoff) could be normally accomplished - i.e. that the landing was a good one. Once the pilot or instructor are satisfied that the airplane has been placed on the runway properly, then the pilot immediately transitions to takeoff and either goes immediately from climbout back into the traffic pattern or takes his or her turn in the circuit if there are too many airplanes using the field for a direct re-entry.
This should not be confused with a go around - which is when an intended landing is aborted - either above the ground or afterwards - due to unforeseen circumstances. A touch and go, however, is good training for the 'emergency case' of a go around. Upon being waved off from landing via radio, light gun or other visible signal a pilot should be immediately comfortable with their next sequence of actions from touch and go practice.
Origins of the phrase
The phrase touch and go also is idiomatic for hazardous or a near thing. This usage long predates aviation; Edgar Allan Poe referred to a character named 'Touch-and-go Bullet-head' in his story 'X-ing a Paragrab' (The Works of the Late Edgar Allan Poe, vol. IV, 1856). Our own Webby notes in touch that the phrase has a nautical origin connotating a narrow escape - when the keel of a ship strikes the bottom but does not take hold, and the ship continues on its way without grounding, this was referred to as a touch-and-go.