(covers information from several alternate timelines)
The structural integrity field (acronymed SIF) was a technology developed by spacefaring species to supplement the natural structural integrity of the material their starship hulls or other constructions were composed of. Engineers used a structural integrity force field to supplement the supports and bulkheads that gave a piece of architecture its shape. This technology was made necessary by the fantastic stresses and probability of a hull breach presented by warp fields and spatial phenomena present in the galaxy.
The field strength was usually measured in percentages. A structural integrity reading of 100% meant that the field was accepting a maximum load of the stresses presented to the structure, and that all physical architecture was safe from that stress. As the percentage approached zero, this meant that more and more of the stress of weapons, gravity, and other phenomena was being experienced by the hull itself rather than the force field. As the hull became damaged or breached, the SIF became less effective as the hull it was designed to hold together started to disintegrate.
When the USS Enterprise-D was trying to break free from the influence of the two-dimensional beings, the structural integrity field stress was exceeding 82 million kilodynes and the attempt had to be aborted. (TNG: "The Loss")
After the USS Defiant was disabled by two Jem'Hadar attack ships in 2372, Worf ordered Stevens to recalibrate the SIF to minimize the strain caused by a torpedo embedded in the hull fired from the Dominion ships. (DS9: "Starship Down")
Later that year, the SIF of Deep Space 9's Upper Pylon Three failed and tore open some of the bulkheads. Several people used the opportunity to steal equipment from the exposed cargo bays. (DS9: "Looking for par'Mach in All the Wrong Places")
In order to stop the USS Defiant from being torn to pieces by exceeding warp nine in 2374, Chief Miles O'Brien took power from the phaser reserves to strengthen the starship's structural integrity field. (DS9: "The Sound of Her Voice")
In an alternate timeline in 2374, the USS Voyager was forced to engage warp drive while the structural integrity field was still down, in order to escape the Krenim weapon ship. As a result, many of the outer hull plates were torn from the ship. (VOY: "Year of Hell")
When the Varro generational ship broke apart due to silicon-based parasites in 2375, Voyager extended its structural integrity field around the ship to buy enough time for the Varro to evacuate the affected areas. (VOY: "The Disease")