For “get you home”, repairs to tubeless tires, there are different types of tire plugs. Some are rubber mushrooms and some are long sticky stringy things. Here’s a nice pictorial of how to use the long sticky stringy tire plugs. It shows a truck or ATV tire being repaired by the looks of it, but the principle is exactly the same for motorcycle tires:
I DON’T AGREE with them that this should be considered a permanent repair though (might be okay for ATVs(?) but not for motorycles). Any tire repair kit you buy will probably come with instructions for use. Remember that you’ll also need something to pull-out / dig-out the nail or whatever and some type of inflation device (CO2 cartridges, mini foot-pump or mini electric pump, etc). Something to clean your hands afterwards is good too!
Tube-type tire inner-tubes can be patched just like a bicycle if you know how to get the wheel off and on safely, but I’d just put a new tube in for safety’s sake; better yet, have a dealer make any permanent repairs to your tires. Any motorcycle dealer should be able to repair tires.
Tubeless tires *can* be internally plugged, patched or a combination of both, but often dealers will suggest replacement of the tire for safety’s sake. Sure, it’s a lot of money to replace a tire, but it’s potentially your life we’re talking about.
Any puncture in the sidewall means a new tire, it’s not repairable.
Be sure you know if your tires are tubed or tubeless though before attempting any repair. My (non-stock but optional for my bike) Metzeler tires say “tubeless” on the sidewall, but I’m running them with tubes. Generally speaking, if you have wire-spoked wheels, then you will have tires with tubes in them (where the spokes terminate at the wheel rim can’t be made airtight so tubeless tires won’t work). Alternatively, if you have alloy wheels, then chances are that you have tubeless tires. Use the RYFM method (Read Your Flippin’ Manual) to find out for sure.
Here’s another link with some great motorcycle tire related topics.
The above should be regarded as informal advice only and I can’t guarantee that the hyperlinks will be there in perpetuity.
You are responsible for your own safety.