Girolamo and Carlo Rinaldi started the Church under the patronage of the Pamphilj family (mid 1650's), but, the patrons did not like the Rinaldi design, and Borromini took over early in the project. Borromini changed the form of the church from the Rinaldis' traditional Greek cross (four equal arms) to an octagonal shape by cutting across the corners joining the arms. The pillars, which were to have been built into the walls were thereby freed and now stand just inside the octagon. Most importantly, Borromini added a tall drum between the walls of the church and the dome and arched the dome much higher, both of which made the church much higher and emphasized the upward sweep and vertical lines that became his trademark. Borromini's plan lengthened the transept arms, and, by using false perspective behind the two side altars, he made them look even longer. The apse, the transept arms and the entrance are all concave, and the other four sides of the octagon have convex architectural elements. Boromini's plan called for stark, white, undecorated walls (as in Sant'Ivo), but Bernini and others later added stone reliefs, statuary, stucco, and paint decoration. By designing a dramatic concave facade (much copied, especially in Germany), Borromini was able to eliminate the forward thrusting monumental staircase, which would have extended to the center of Piazza Navona. He added convex extensions on each side of the facade over parts of the fabric of the Palazzo Pamphilj to achieve a broader front that matches the increased height of the dome. The twin towers above the convex extensions (which are criticized as too high) were added later by others. The dome exterior, incorporating the lantern designed by the Rinaldis, is usually ascribed to Borromini. The Fountain of the Four Rivers, directly in front of the Church, is, of course, by Bernini.
An aerial view of Piazza Navona with the Church of Sant'Agnese dominating: http://www.roma2000.it/znavona.html
400 Years of Borromini Internet site: http://www.borromini.at/e/index.html
The splendid Morlacchi drawing of the facade of the Church http://www.bestitaly.net/morlacchi/01b.htm
The Borromini biography from the Catholic Encyclopedia: http://www.newadvent.org/cathen/02688d.htm
PS -- There are many contradictory accounts of the martyrdom of St. Agnes of Rome. The following, much favored by local guides, is among the oldest and most lurid. Agnes, who was only 12 or 13, was said to be the object of the desires of a Roman prefect, but she spurned him saying that she was already committed to holy virginity. The prefect and his friends first spied on her and then dragged her off to a house of ill repute where they publicly stripped her. Her hair grew miraculously to protect her modesty. She was then set alight, but the fire miraculously left her and consumed the prefect instead. The prefect's father arrived at the scene and demanded that she be killed. The crowd that had gathered agreed, whereupon Agnes miraculously brought the soldier back to life, thereby converting to Christianity both him and his father. Despite the pleas of the soldier and the father (now on her side) the crowd would not relinquish its victim, and she was, after great suffering, decapitated. Her head was kept in a shrine on the spot where her church (in Agone) now stands. Other accounts tell of a more orderly procedure in which Agnes was accused of being a Christian, brought before a prefect, exposed (and miraculously covered by her hair), burnt at the stake (and miraculously protected) and finally beheaded. One account speaks of a man who was blinded after catching a momentary peak at her nudity -- Agnes cures his blindness. The church of Sant'Agnese fouri le Mura, on Via Nomentana, was built in her honor at the site of the burial of her body. (Her detached head stayed at Sant'Agnese in Agone.) Accounts of her martyrdom place her death either in the persecution of Diocletian (about 304 AD) or in one of persecutions of the previous century.
The Catholic Encyclopedia on St. Agnes of Rome: http://www.newadvent.org/cathen/01214a.htm
PPS -- the "in Agone" in the name of the church has nothing to do with the agony suffered by Agnes. The "agone" were nude Greek athletic contests of the type for which the Stadium of Domitian, the current Piazza Navona was built. Domitian dreamed of reviving the Greek style games and de-emphasizing the bloody gladiatorial contests. But after only one disastrously ill-attended season, he switched his stadium to the ever-popular gladiators. The name of his "agone" stadium stuck, however, and so Agnes was beheaded and her church was built "in agone". And "in Agone", from the name of the Church, was gradually corrupted until it became the "navona" of the present name of the Piazza.