A diagram of the fallopian tube joining the ovary (left) to the uterus (right), courtesy Gary S. Burger, MD, via Wikimedia Commons
The fallopian tubes are hollow, cylindrical structures that branch from the feudal ends of the uterus and end near the ovaries. They're used to transport ova from the ovaries to the uterus.
An ectopic pregnancy is dangerous complication where a fertilized eggs embeds in the fallopian tube instead of the uterus. Once the egg grows to the size of a fetus, it will break through the tube, causing bleeding in the abdomen which can lead to death within hours.