The Picard Maneuver involves warping from one location to another, within the enemy's visual range, then firing all weapons in an alpha strike. The quick warp jump enables the attacker to outrun light from it's starting position to a closer position; light from the original position is still arriving at the target when the attacker reaches its new location, creating the illusion of the attacker being in two places at once. The target's weapon systems -- presumably relying upon lightspeed sensors -- remain locked on the attacker's original position, and the humanoid crew of the target can't respond quickly enough to redirect fire to the attacker's new location.
Unless the target knows in advance what is happening, it will probably waste its weapons fire on the image of the ship rather than the true ship. If the attacker's alpha strike is sufficiently powerful, it can quickly overwhelm the target's shields, possibly destroying the ship.
The Picard Maneuver was not widely used in the Dominion War, possibly because it is more advantageous to divert warp power to shields and approach heavily shielded. In TNG "Hero Worship", Geordi diverted fusion power to the shields, resulting in a doubling of shield strength; directing warp power to the shields would provide an even greater increase.
The Picard Maneuver appears to rely on a tactical officer's inexperience in combat and slow human reflexes. A competent tactical officer could theoretically program his console to automatically target and attack any enemy ship that suddenly appears on its sensors, removing much of the benefit of this maneuver. Such possible pre-programmed responses are evident in TNG "The Wounded" when the Enterprise-D raises its shields automatically in response to a Cardassian sneak attack, or The Final Frontier where the Enterprise-A detected an incoming Bird of Prey but was not programmed to automatically raise its shields.
The success of this tactic depends on the inability of the target to follow the faster-than-light movement of the maneuvering ship. This tracking limitation implies that at combat ranges, warships rely heavily on lightspeed sensors to track other vessels and verbal orders to direct fire.
As far as we know, the Picard Maneuver has only been successfully used once. Captain Picard, then captain of the USS Stargazer, used it to counter-attack a Ferengi ship that ambushed the Stargazer, destroying the attacker in one salvo. As noted above, starship captains seldom use this tactic, since it would typically be more advantageous to use the same power to strengthen the ship's shields. The Stargazer's shields had already been depleted in the initial Ferengi assault to the point that diverting extra power to them would be useless, but the warp engines were still in good condition, leaving this maneuver as his best option.
Under the influence of a Ferengi mind-control device, Picard attempted to repeat the attack against the Enterprise, but knowing what to expect, the Enterprise crew were able predict where the Stargazer would drop out of warp by detecting the compression of trace hydrogen ahead of it, allowing them to immobilize the Stargazer with a tractor beam.. Since they had experience with the maneuver, they knew that the second, "new" image would be the correct one.
The Picard Maneuver is occasionally referenced as an example of warp strafing, but this maneuver specifically involves dropping out of warp before firing weapons.