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When your gourds have reached their full size, the vine they're growing on will start to die off on its own. At this point your gourds are ready for harvesting, but you'll make the job a lot easier on yourself if you leave them to cure on the vine. Give them several weeks to a month for the curing process to occur; as you check in on them, you'll notice them getting lighter and lighter. Unless you notice animals and bugs eating the gourds, there's no fear of rotting or going bad. If you're harvesting edible gourds, they'll need to be removed from the vine when they're still young. If you have to cut the gourds early, wait till the vine at the top of the gourd has turned completely brown and dry. Turn the gourds occasionally and move them around to keep them from touching. The curing time varies from gourd to gourd depending on its size (and therefore water content). Check the gourds on a weekly basis to tell if they're ready. Feel the skin and check the firmness of the gourds; if they are at all soft or squishy, they are rotten and should be thrown out. When the skin feels hard and slightly waxy to the touch, they are likely ready to be cut. Shake the gourd as the final test to see if they are fully cured; if they are ready, they'll sound like a rattle with the seeds banging around on the inside. Use a pair of scissors or shears to cut the gourds from the vine. Although it is not required, you can treat the shell of the gourd to change its appearance and to help it last longer. Wash the gourd with a bit of dish detergent and warm water to kill off any bacteria. You can then use a bit of sandpaper or steel wool to shine the outside of the gourd, and add a layer of wax or shellac to finish off the shine. You can decorate gourds by painting the outsides as well. Your gourd will last for many years with the seeds inside, but if you would like to save the seeds for the next year's planting, you may do so. Cut the gourd open to remove the seeds from the inside. Follow the same process of propagating the seeds (as aforementioned) to help speed up their growth. You can keep the shell of the old gourd, and you'll have the seeds to create plenty of new gourds as well.

Summary:
Leave the gourds to cure on the vine. Remove the gourds. Treat the shell of the gourd. Consider saving the seeds.