Hi and welcome!
An airstone is like a porous little rock (or big, depending on what size you get) that hooks up to an air pump, via airline tubing (just silicone tubing). When the pump gets turned on, air is forced through the tubing to the stone, pushing bubbles of air into the water (provided the airstone is in the water).
In most cases, airstones are not necessary for tanks. An air pump and airline tubing (and sometimes an airstone) are used to run under-gravel filters, at least in the little one-gallon AquaView tanks from Wal-Mart, and sponge filters.
Some people like the look of all the bubbles in their tank and have what is commonly referred to as a "bubble wall". To achieve that effect, a long airstone is hooked up to the air pump and tubing.
If more surface disturbance is needed (to help keep oxygen levels in the water higher), an airstone is a good idea. I had one in my tank when I had the temperature at 86* to treat ich. The warmer water gets, the less dissolved oxygen it can hold. The airstone helped put more oxygen in the water and create more surface disturbance to allow some more oxygen in.
I don't know if there are airstones that don't require a tube and pump, although I would doubt it. You can get tablets that will release oxygen for some period of time (not sure what the time period is, as I have never used them) which are sometimes used when transporting or shipping fish or plants.
The majority of Bettas will do just fine in a five-gallon tank. There may be the odd one or few that would prefer something a little smaller, but for most, a five-gallon would be great. A filter is fine, too. If the filter creates a lot of current, the Betta might not like that. Make sure there's a place in the tank with slower-moving water. Some Bettas likes airstones, some don't. Some like to play in the bubbles it produces. I don't know about snails liking airstones, as I don't know much about snails. If it's a five-gallon tank, you ought to be able to have an airstone in there, if you want, and still have some space for the Betta to get away from it, in some "calm" water. I believe the majority of Bettas prefer slow-moving water.
I hope that helps you out some
If you have more questions, feel free to ask!
Oh, depending on the temperature of the room you're going to have your Betta in, you may want to get a heater for his tank. Temperature fluctuations of more than a couple degrees can stress the fish out and make it more prone to getting sick.
Good luck
Pamela
aka Married Lizard