First of all, the umbilical cord is long, usually around 55 centimeters or two feet. This offers a lot of slack so that a baby can move quite freely without danger. True, there is the chance of entanglement, but things are so loose that there is seldom constriction of the cord. Adding to the safety is the sheer volume of amniotic fluid, which keeps the cord from being pushed up against the wall of the womb. In fact, some instances of fetal distress during labor have been treated successfully by instilling fluid into the womb to restore the volume lost after the "water bag" broke.
A cord wrapped around the neck happens often, but because of the reasons above is seldom a problem. If the cord is wrapped around two or more times, however—much rarer—there could be a constriction of one of the loops. This can usually be picked up while monitoring labor, with subsequent steps taken for safe delivery.
The cord has a specialized membrane that plays a role in fluid exchange and other important interchanges. Contained within it are two arteries and a vein. These are the main channels for funneling oxygen and nutrition to the baby. Protecting the blood vessels from being crushed by twisting or compression of the cord is a substance called Wharton's Jelly. It runs the course of the cord, surrounding the vessels, acting as a cushion to keep the vessels from kinking. Indeed, a baby whose cord has little Wharton's Jelly is at risk for fetal distress. Thankfully, this is usually an incidental finding after a normal delivery. Also, the regular pulsations within the cord, the constant forceful flow of blood, tend to act as a "straightening" phenomenon to prevent twisting.
True knots in the cord are very rare, and are most often seen as another incidental finding after a safe delivery. Lifting arms above one's head doesn't cause knots, as many patients have heard. This is a myth. A true knot is usually not tight, and the Wharton's Jelly generally does what it does best—protect the all-important vessels.
Cord entanglement among twins, however, is a real threat if they share the same sack. This type of twin pregnancy is usually the exception, most twins' cords being separated safely by a membrane between the two babies. But it is important to use ultrasound as early as possible in the case of twins to demonstrate that the membrane is there.