You would need five things right off, before you even dismount the tire. This is assuming you have the equipment to dismount the tire and or tube from the rim and remount it after you are done. 1. patch. These are offered in multiple sizes and shapes, usually round or oval. Most auto supply store either have them or can order them in for you, or you can have them delivered by online order. 2. Vulcanizing cement. The brand I find is made by an outfit named Group 31. They sell through OreillyAutoparts and others. Depending on where you buy it, you can even get it in a five gallon drum. 3. Abrasive. Most of the tire kits I saw when I was a wee bit younger had a little metal disk in the kit, or the lid was perforated like the thing you use to shred potatoes when making hash browns. 4, Roller 5. Clamp and (maybe) pads Now you have these things gathered up, how big is the damaged area? The method I described below is for something simple like a nail puncture or similar damage.
Once you have the tire or tube dismounted from the rim, you air it up until you can find the puncture. Let the air back out until fully deflated. Rough up but do NOT shred the rubber at the site of the puncture.
Most of the patches that you buy have a piece of plastic film on one side. That is the side that you place over the puncture. Apply some of the vulcanizing compound over and around the puncture, then peel off the protective plastic from the patch and, centering the patch over the puncture, apply it where you placed the adhesive or vulcanizing compound. Using some sort of roller, roll over the patch to ensure any and all air bubbles are out from underneath it. Then, grab your clamp and (if needed) your pads, and apply them to hold the patch in place until the adhesive cures. If you don't have the patches, I think you might be able to do the same thing with a clean , dry section of inner tube you might have laying around. Check bicycles if you don't have one for a car or truck. You shouldn't need much, depending on the size of the puncture,
Most of the vulcanizing kits I grew with, you actually used a match or lighter and set the adhesive aflame to truly vulcanize the patch to the parent material. With the commonality of plugs nowadays, it has been years since I have seen that done.
As a side note, as a kid, we used to patch our bicycle tubes and tire with either duck or electric tape or ride on the rim. It simply wasn't a good use of a tire kit when they were needed to keep other more productive things (Like the old Ford 9n tractor) going.