To redpoint a climb is to successfully free climb (i.e., not aid) a route without falling after a previous unsuccessful attempt.
So let’s say I head up to Goober Route, a local sandbagged 5.8. I go for the onsight (trying to climb it without any previous knowledge of the route and without falling) but fall above the third bolt. Not a big deal. I figure out the right sequence and eventually make my way to the top of the route. I rappel down to the base and pull the rope, eager to try again and climb the route cleanly. I do, successfully navigating the crux at the third bolt, and let out a shout of joy once I clip into the anchors. Congratulations to me! I have now successfully redpointed the route!
So where does the word redpoint come from? The Wikipedia entry says the following:
“The English term “redpoint” is a loan translation of the German Rotpunkt (point of red) coined by Kurt Albert in the mid-1970s at Frankenjura. … Once he was able to free-climb [a] route, he would put a red dot at the base of the route.”
This jives with what I heard legendary Yosemite/Joshua Tree/all over climber/free soloist John Bachar (who later tragically died in a free soloing accident) say at a slideshow a few years back, but I want to add a bit more context. The way I heard it, climbers in the Frankenjura would take nail polish and paint an open red circle at the base of a route that was being attempted but hadn’t yet been successfully climbed clean (i.e., without falls or aiding). When the route had been successfully climbed, the climber would then fill in the circle so that it was solid red dot or, in other words, a redpoint.
Bachar went on to say that he brought the word redpoint back to America and popularized it here in this country. I don’t know if that’s really what happened, but that’s the story.