A personal story is a set of two cards detailing a goal important to the specific investigator and the results of success/failure in pursuance of that goal. The cards also expand on the background story of each investigator, tying their background into the goals on the cards. In many cases, the personal stories follow up on loose ends from the original backgrounds of the investigator cards. Personal Story cards first appeared in the Innsmouth Horror expansion.
The first card details the goal and the conditions of success and failure. The second card details the consequences of success or failure. Once you succeed or fail, you enter this second card in play with the appropriate side up. This effect stays with the Investigator throughout the game (once you succeed, the consequences of failure do not enter gameplay even if you later fulfil the conditions; and vice versa).
For the personal stories themselves, see the individual Investigator entries.