Star Trek Canon Special Note
Not everything listed in the origin is Canon. That is because only television shows and movies are considered canon. This being a comic website there are many things here that are not canon. So unless it is contradicted in film it should be listed as part of the character biography.
Number One became the first officer of the Enterprise under Captain Christopher Pike. She is a very logical person and a very capable officer. She was even offered to command the U.S.S. Nelson but turned it down due to her loyalty to Captain Pike.
At the time, in real life the term "Number One" was a common nickname for whoever was second in command of a ship. Thus, calling her "Number One" instead of her name is not unlike referring to a physician only as "Doctor" or referring to a university instructor only as "Professor". Roddenberry referred at the time to the character as having a mysterious past that would intrigue viewers as the series went on -- however, because of executive meddling, she appears only in the pilot and in the two-parter that was created from the pilot, "The Menagerie".
The Character of Number One (played by Majel Barrett) was created in the first ever pilot of Star Trek called the Cage. When the Pilot was viewed, executives decided that many changes had to be made. In particular, they insisted that the television audience of that time (the early 1960s) would never accept a female second-in-command.
Executives wanted to get rid of both Spock and Number One. Gene Roddenberry the creator of the show knew he could only keep one so he kept Spock and married Majel. This meant that the character of Spock changed as well. Spock became much more logical like Number One than he was in the first pilot. Majel did come back to the show when the second pilot was picked up as the character of Christine Chapel.
Not much was done with the character of Number One after the pilot. The show was split into two episodes called the Menagerie. It was not until about three decades later that the character really grew. In the comic series Star Trek: the Early Years Number One goes on many adventures. She even gets command of the Excelsior and has Chekhov at the helm. The most notable change in the character was what her name was. The television show only referred to her as Number One.
In some comics that is her actual name. In others it is Eunice Robbins. According to Gene Roddenberry and the creators that came after only the television shows and movies are considered canon. So comics and novels can play with these ideas but as far as the shows are concerned her name is Number One.