There are a couple of techniques you can use depending on whether you’re dealing with a hole or a tear. If there is no obvious rip and the slide is leaking air, but you can’t find the location of the hole, then you can find it using the same method you would use for a punctured tire: make a solution containing water and washing up liquid and pour or spray it on the inflated slide. If bubbles appear anywhere on the surface, then that’s where the hole is. Make sure the area is completely dry before you repair the hole using a puncture repair kit (often supplied with a new slide, or you can get them in your local hardware store).
For larger areas where the vinyl has actually ripped, temporary repairs can be made with duct tape – it might not last long, but could be enough for your kids to finish their pool party so you can make a more permanent repair later. To do the job properly, you can buy a rubber ring, cut it up to make a patch the correct size for the rip, allowing plenty of overlap around the edges, and glue it in place with waterproof epoxy glue.