The air talisman is an item that allows players to enter the air altar. At the altar, the player can use the Runecraft skill to turn rune essence or pure essence into air runes.
Also at the air altar, an air talisman can be combined with a silver tiara to make an air tiara for 25 Runecraft experience by using the tiara on the altar while having an air talisman in the inventory. This consumes the talisman, but the air tiara can be used the same as an air talisman and can be worn, opening an inventory slot to carry another essence.
Air talismans can be used to create combination runes. The talisman is consumed when used as the "secondary" talisman to make these runes.
Air talismans can be obtained as a reward from the Rune Mysteries quest, or as drops from various monsters. The lowest levelled monsters to drop them are level 2 goblins, but they are not a frequent drop. The level 13 Air wizard located south of Falador occasionally drops air talismans. There's a 1/20 chance of obtaining one from them. Level 86 Abyssal leeches often drop talismans, including air talismans, but the leeches are in members areas and the areas are dangerous for lower-levelled players. You can also get a drop of 25-75 noted air talismans from Dagannoth Prime.
For an exhaustive list of all known sources for this item, see here