One you become a mathematician, you will work either in applied mathematics, or theoretical mathematics. Sometimes, theoretical mathematics is referred as pure mathematics.
Gauss, Einstein, and Cauchy were great mathematicians. Gauss is probably the greatest ever.
There is something common with them. They all conduct research extensively.
Those who work in applied mathematics try to learn more about what math has to tell us or how math can be applied in almost any areas of life, such as economics, social science, engineering, biology and many others.
This means that they must find mathematical models that can be used to solve real-world problems.
Those in pure mathematics just want to know more. They are the one who are going to come up with new axioms, definitions, and theorems.
Then, they will devote their lives in proving those theorems. Throughout history they have given us many great achievements that have had profound effects in engineering, science and many other discipline.
Needless to say that one needs to be very good at math to become a mathematician.
Above all, a passion and a love for math are a must. I know some folks who are good at mathematics, yet they certainly do not want to do it everyday.
Mathematicians do math everyday and they are expected to come up with solid proofs for theorems and good mathematical models of real-world problems, and it can be rewarding when one has discovered something.
However, it can be very frustrating when one is looking, but has not found a solution.
For instance, no one has proved the twin prime conjecture yet(there are an infinite number of twin primes)It has been proven though that there are an infinite number of prime numbers.
Getting the job:
Many jobs require advanced degrees, such as a master, or PhD. Some jobs ask only for a bachelor's degree.
For instance, to work for the government, you must have a four-year degree with a math major to get an entry level position.
Recommended classes are calculus, linear and abstract algebra, and differential equations.
If you also know numerical analysis, topology, and probability and statistics, then it is a plus.
There are also entry-level positions available in private sector. You will not be called though with the title; you will be called a programmer, systems analyst, systems engineer, or researcher.
Most jobs are found in industries such as government, education, manufacturing, and the private sector.