Spirit Gum
Spirit gum is one of the cheapest, and is an all-around favorite, for those who will have on their appliances for a short time. Simply apply it to the piece, let it become tacky, and apply to where you want it. However, it's not waterproof or sweatproof. In heavy sweat, it's been known to crystallize, and that doesn't feel too good against your skin.

Prosthetic adhesive/pros-aide
Next is prosthetic adhesive, or pros-aide. This is a milky white substance, that is similar to what Lyonshel Studios supplies, if not the same thing, with their appliances. You apply it to the appliance, and possibly to yourself as well, let that dry, and add a second coat. When that's dry, carefully put the appliance on you. This is not the easy part, since it will stick to you, no matter what.

Medical Adhesive/surgical adhesive
Medical adhesive, also known as surgical adhesive, is the favorite among makeup artists, as it's quite strong. It's known to outlast the wearer. When boogi wears his Mutant Cat, it's not unusual for him to wear it for 15 hours straight without a touchup of the adhesive. It's that strong and long-lasting. It's also the most expensive, going for around $18-25 for a fluid ounce, with a 4-ounce bottle around $60-65. Kryolan makes this.

Telesis
There's also Telesis, which is like medical adhesive. This is more resistant to water and sweat, though not as strong as medical adhesive. Application is the same as above. This has several versions.

Liquid latex
Yes, liquid latex can be an adhesive, though after several hours, will start to come off on its own. When used with foam latex appliances, actually makes the appliance stronger, due to the latex buildup.

superglue/cyanoacrylates
Let's face it… We all love this stuff for various reasons. How many of you have glued your fingers together or onto something, or worse? This is NOT the stuff to use for putting appliances on your face unless you want to have it on for a long while til you shed enough skin for it to come off. This, folks, is NOT a pretty sight.

other adhesives?
A medical supply house may have something you can use, but as always, ask questions about the product. There are variations of adhesives that will come off with water, to some that won't come off til you've shed some skin.