One of those obscure climbing terms used in rock climbing, specifically sport climbing.
To red point a route, the climber can try the route as may times as (s)he wants. (As opposed to on-sighting or flashing a route, which are first-try things). The climber can discuss the route and watch others attempt it. For the red point, the climber must climb the route from the start (usually the ground) leading (see The Sharp End) and placing and clipping quick draws as (s)he goes to the top without falling or hanging on the rope.
There is often some sloppiness in terminology, especially among sponsored climbers, between red pointing and pink pointing. (Pink pointing is the same thing, but with the quick draws already clipped to the bolts, which makes things quite a bit easier.) As far as I'm concerned, as long as you're clear about what you did, it's fine with me. (e.g. "I stick clipped the first bolt, and the second and third draws were in place from Bob's last attempt, but I placed the rest and got the, uh, red point, kinda.")