So, I don't really know all that much about mountain bikes and how to fix them up. I just started riding my bike a lot. I got a nail puntured through my rear tire...went in through the tread and out the sidewall. I've got 26" Tioga Psycho II tires and was wondering if (this may sound dumb) they use an inner tube (I'm assuming they do), and if so, once I replace the innertube, can I still use the tire itself, or is its days numbered??

Thanks in advance...I know it's a dumb question but your help is greatly appreciated

It is possible to set up a tubeless bike tire but its very uncommon. You have innertubes.

If the tire is only punctured and not torn just patch or replace the innertube.

so, how 'torn' are you talking? there's a single puncture hole on the tread and then there's a hole with a small 1/8"-1/4" gash in the sidewall. Is that going to cause a problem?

The single puncture hole is unlikely to present a problem, the gash might or it might not. Once you get a new/patched tube in there and pump it up just check if it bulges around the gash which would indicate damaged belts. If so then the safest option is to replace the tire, if it doesn't bulge then it should be OK.

Remember that it's the belts embedded inside the rubber which give a tire its structural integrity, not the rubber itself. Also, you should see the recommended maximum and minimum pressures printed or embossed into the sidewall, stay within that range. Easiest way is to get a pump with a built in pressure gauge. I've owned and used dozens of different pumps but the only one that I like and now have is the "Road Morph" made by Topeak. It mounts on your frame, has a built in gauge, and works like a floor pump.

I feel OK with nail holes in my tyres. I get worries once I get a slash through the fabric that is longer than about 5mm.
Inner tubes can be patached pretty easily although most of us carry a spare tube for roadside replacement and patch in comfort back home.

you might also put a piece of fabric (or maybe a product exists for this) against the sidewall gash to add some structure there.

To quote Sheldon brown:

Boot
A temporary patch to cover a large hole or cut in a tire. A boot does not need to be air-tight, but needs to be sufficiently strong and stiff to prevent the inner tube from bulging through the hole in the tire.

Many things may be used to boot tires, including leather, currency, food wrappers, etc. Patches intended for tube patching will not usually do a good job, because they are stretchy and will blow through the hole just as an inner tube would.

The best material for boots is a piece cut from an old thin-tread tire, preferably a tubular. Many cyclists carry such a piece of tire for this purpose.

A boot does not need to be glued in place. It will be held in position by the pressure of the inner tube.
Bolt-on