An anesthesiologist makes sure that a patient is alive and comfortable during/after surgery
. This may or may not involve putting the patient to sleep.
A crazy person with a knife (surgeon) may be cutting into the patient, who may be too asleep to even breathe for themself. It is therefore the anesthesiologist's job to act as the medicine representative in the operating room and keep the patient well and pain-free. This can be done via intravenous medications, inhaled agents, and locally-injected drugs.
A anesthetist is able to perform most of the same functions as an anesthesiologist (just as a nurse practitioner is trained to function as an internist). The best hospitals have a mix of anesthesiologists and nurse anesthetists, backing each other up.
Due to effective research in the field, anesthesiology has become far safer today than it was even 20 years ago.