In "The Name of the Doctor", Clara threw herself into the Doctor's timeline on Trenzalore to stop the Great Intelligence from sabotaging it. Entering the Doctor's timeline splintered and scattered her across the Doctor's entire life.
As a result, a person with her appearance would show up every so often in his lifetime and intervene in some way to save the Time Lord's life. Though each splinter-Clara never knew about her other selves or about the Doctor. The Doctor rarely noticed her. A complete account of Clara's echoes has never been made, but a few exceptions to this included an echo that met the First Doctor and guided him towards taking the correct TARDIS, she was noticed by the Third Doctor while he was driving Bessie, the Oswin Oswald echo communicated directly with the Eleventh Doctor, and the Eleventh Doctor met and initiated a friendship with the echo known as Clara Oswin Oswald.
The only Doctor Clara did not meet during this time was the War Doctor, as the Doctor was able to keep his existence secret from the Great Intelligence; this omission was rectified in "The Day of the Doctor". She later went on to travel with the Twelfth Doctor and the story "Blood and Ice" (comic story) saw Winnie Clarence, another echo, assist the Twelfth Doctor, confirming that Clara's echoes were not restricted to helping the first eleven "official" Doctors.
In "Day of the Doctor", she also met a future incarnation of the Doctor known as the Curator (the Titan Comics comic book series confirmed the Curator as a future incarnation of the Doctor).
It is not known whether Clara remembers any of her echoes. All that is known canonically, as indicated in both "Day of the Doctor" and "Blood and Ice" is that she has vague recollections of seeing other Doctors prior to the Eleventh Doctor rescuing her.