Can I root my stock-Gingerbread Captivate without flashing a new kernel?
Unfortunately for us (not so much for manufacturers and carriers,) Gingerbread closed the exploits that were previously being used to gain root privileges on older Android OS versions.  This means that tools like SuperOneClick and similar will no longer work.  The only way to get root on Samsung devices with Gingerbread is to flash a pre-rooted image onto the phone with a tool like Odin or Heimdal.

Forums like XDA-Developers and Rootzwiki contain detailed instructions for flashing such an image (kernel only, full stock system, or a completely custom ROM.)