Under general anesthesia, the muscles supporting your airways will relax and a breathing tube will sometimes be put down your throat to ensure that you get enough oxygen. The relaxing of the muscles in the airways also relaxes the muscles that prevent stomach acids or food from crossing into the lungs. This is one reason why you are required to fast for about 8 hours before the surgery begins. Although it relaxes the muscles involved in breathing it does not completely stop the breathing so you do not literally stop breathing. However, your breathing does become weaker and you might breathe in less oxygen which is one of the reasons that the tube is there.
Another purpose of the tube, however, is to protect the lungs from stomach acids. The anesthesiologist will use the breathing tube to monitor your breathing. He or she will also monitor your fluid levels and other vital signs. The breathing tube is not absolutely required but it does greatly increase the safety of the operation.