The word typist originally refered to a person who operated a typewriter (a device difficult to find today). However, now the term refers to anyone capable of typing, usually on a QWERTY (though sometimes DVORAK)-style keyboard.
In the professional world, a typist is someone who is employed for the primary purpose of entering date into a computer (modernly refered to as data entry).
Before the days when a computer could be found at nearly every desk or in every cubicle, any number of people were hired to work as typists, producing letters, reports, and even punchcard programs for their employers.
Today, secretaries, administrative assistants, and receptionists are more likely to be found than typists. While these individuals still perform the same tasks as typists, they also have various additional duties.
A typist's skill is measured by a number of benchmarks including WPM or words-per-minute, accuracy, and less often CPM (characters-per-minute) or GWM (gross words per minute).