A Talisman is attuned to the Runecrafting altar after which it is named. For example, an air talisman is attuned to the air altar. A player can use the talismans to reach the altars via the 'mysterious ruins' found throughout Gielinor. Players cannot use the talismans until they have completed the Rune Mysteries quest, but it is perfectly possible to acquire talismans, from drops or trading, prior to starting the quest. A talisman can be used repeatedly to enter its altar and is never consumed for this. The talisman only grants access to its altar; the player must have the correct Runecrafting level to craft runes there.
Talismans are also used to make combination runes and are consumed in this process.
A talisman can be bound with a tiara, creating an altar-accessing item that is worn on the head rather than carried in inventory. All talismans except the elemental talisman can be bound in this way. When you bind a talisman with a tiara it gives the player quicker access to the altar (allowing the player to left click to enter).
Note that players using the Abyss to reach the runecrafting altars do not require talismans (or a tiara).