Pitocin, which is a synthetic form of the hormone oxytocin that plays an important role in labor and birth, is the most commonly used drug used to induce labor in the United States. Labor induction with pitocin is generally quite effective, though it can be used in combination with methods like stripping a woman's membranes or breaking her bag of waters too. Pitocin is also the most likely drug you will encounter if you have a stalled labor. Many hospitals have specific guidelines on how long a labor should last before they introduce interventions.
"Failure to progress" refers to a situation in which a woman's labor is not progressing at the rate the hospital feels is optimal, or even to cases in which labor totally stops. Many hospitals have adopted the policy that the baby should be born within 24 hours after a woman's waters have broken, for instance. If your labor has stopped or contractions are not getting any closer together and your cervix is not dilating (or is dilating more slowly), labor augmentation with pitocin is a common procedure. An augmented labor will be speedier than a natural one, or at least that is the aim.
In some cases, augmentation is definitely warranted. It is true that the risk of infection grows as time passes when the bag of waters has been ruptured, especially with frequent vaginal exams. If you are not comfortable receiving pitocin for this reason, discuss this with your healthcare provider in advance. You may choose a doctor who is more open to natural childbirth, by asking questions about when they tend to augment labor as well as about other routine interventions. Pitocin is not often a good idea in cases of fetal distress, since the drug itself can cause contractions so close together that fetal distress can in fact be caused by it.