An anesthesiologist is a doctor who specializes in managing pain and taking care of patients while they are getting ready for surgery, in surgery, and recovering from surgery. They do not perform surgery themselves, but they make sure that patients are safely put to sleep and/or anesthetized (which means they can't feel anything) during surgeries. They also make sure patients' vital signs are stable during surgery and help stabilize the breathing of patients who are not in surgery but are very sick and need a ventilator. They also manage long-term and short-term pain for patients, mostly by prescribing drugs but also by doing minor procedures.
Anesthesiologists make a lot of money. Reported averages vary widely, but most make anywhere between 200,000 and 400,000 dollars per year. You should be aware though that most doctors in most different specialties have salaries in that range as well. Doctors in general make a lot of money.
To become an anesthesiologist, first you have to become a physician. In America, first you need to earn a college degree, and while in college you need to take a set of courses that are pre-requisites for medical school; these include biology, chemistry, and physics. You'll need to take the MCAT, which is like the SAT for medical school, around that time as well. Then, you will go to medical school for 4 years, graduate with your MD or DO degree, and begin a residency in anesthesiology. Residency is the time after medical school when doctors learn to practice their specialty of choice before they are allowed to work on their own. For anesthesiology this requires an additional 4 years. That makes the minimum time 12 years from high school graduation to becoming an anesthesiologist, unless you graduate college early.
Edit: Based on the way you spelled "anaesthesiologist" I guess you may not be from the US unless you just googled the spelling. I have no idea how specifically to become one in other countries, but no matter where you go you must become a physician first.