Warp factor one, or more rarely, Warp one, is believed to be the speed of light. Starships using warp drive can achieve this speed, but it is ridiculously slow for interstellar travel. Still, it was the velocity ordered almost exclusively by Captain Kirk in all of his missions between 2265 and 2266. It is believed that Kirk was either told by his chief enginneer that the ship was incapable of going any faster, or that the brash young captain may have failed the warp mechanics course at Space Academy.
Whetever the reason, the entire crew was out of contact with the Federation for several years due to the time dilation effect. It was only the fortuitous events over LaPig which caused Spock to make up a new formula for cold fusion and return Enterprise to the correct place in the time-line. (TOS: "Time Travel Invented in Last 30 Seconds of Ep; Never Mentioned Again".) It was especially fortunate for Kirk, as he had engaged in a number of liasons over that period of time and thus avoided any number of paternity suits due to the fact that the children were never born.
In the 24th century, as a mark of respect to Kirk's accomplishments, it was decreed by Starfleet that the flag ship of the Federation could never go faster than warp factor one. Fortunately, another incredible –and vaguely gratuitous—spatial anomaly (this time, the Lexus), removed that nonsensical general order. Star Trek: What Women Want.