I am not a doctor.
In popular culture, a woman's water breaking1 is indicated by a sudden discharge of water-like fluid from the vagina, and indicates that the baby is ready to come out. This is generally correct, but is often exaggerated on TV and in movies. Only about 10% of labors start with the breaking of the water, and it is often a trickle rather than a gush.
In the womb the fetus is protected by a sac filled with amniotic fluid. When a woman enters labor this sac ruptures, releasing the fluid. In many cases this fluid is released in a rush, but in other cases, particularly if a woman is standing2, it may be released more slowly. The breaking of the amniotic sac also triggers the uterus to increase production of prostaglandins that stimulate labor; because of this, breaking water is often seen as an indicator that active labor is beginning, and sometimes doctors will artificially break the sac in an attempt to speed up a slow labor3.
In some cases, a woman's water may break considerably before labor starts. This can result in an increased risk of infections, and if labor has not begun within 24 hours of your water breaking, or if your water breaks before the 37th week of pregnancy, you should consult with a doctor. Other signs of possible problems are if the fluid has a greenish color (indicating that there may be meconium in the womb), if it is bloody, or if it smells bad (indicating an infection). However, it is also important to remember that your uterus puts increasing pressure on your bladder as the pregnancy progresses, and Braxton Hicks contractions can cause some leakage of urine; this can sometimes be mistaken for water breaking.
It is possible for a baby to be born without the amniotic sac rupturing, but this is unusual. If the sac does not rupture on its own the doctor or midwife will often pierce it4. However, babies can be born fully enclosed in the amniotic sac, and are none the worse for it. This is sometimes called a cauled or veiled birth.
Footnotes:
1. A note on usage: you will almost never hear the exact phrase 'water breaking' except in articles talking about someone's water breaking. 'break' is conjugated as it would be were it not part of a phrase, resulting in either a separable phrasal verb ("my water is breaking" or "her water will break"), or an irregular past tense form ("my water broke"). Adverbs will also be placed just before the 'true' verb ("my water just broke").
2. When the mother is standing, gravity pulls the baby down so that its head blocks the flow of the amniotic fluid. When the mother is sitting or laying down, this effect is diminished.
3. While you usually hear about an 'artificial rupture of membranes' (ARM), AKA an amniotomy, there is another, simpler procedure that a doctor will often try first, called 'sweeping of the membranes' (or sometimes, 'stripping of the membranes'), in which she will simply run a finger around the inside edge of the cervix; just moving the membrane off of the cervix will sometimes stimulate enough prostaglandin to start labor; this only works in about one out of eight cases, but is very simple and carries no risk. ARM should only be done once the baby's head is descended into the pelvis; otherwise you may risk the cord slipping around the baby's neck.
4. There is some debate over whether the sac should be artificially ruptured in a healthy pregnancy and labor. If there are no signs of complications, medical professionals may prefer to leave the sac intact, and there is no harm in doing so, if labor is proceeding apace. From what I have read midwives are more likely to leave sacs intact, and doctors are more likely to pierce them; however, if you have a question you should ask your medical professional what they prefer.
References:
MayoClinic: Water Breaking
MidwifeThinking: In Defence of the Amniotic Sac
eMedicineHealth: Labor Induction and Augmentation
WebMB: Rupture of the Membranes
American Pregnancy: Inducing Labor
Home4Birth: Premature Rupture of Membranes