Holding it in one's possession and then chewing it like bubblegum releases oxygen, allowing the person in question to act in thinner atmospheres as if under Earth-like conditions. Oxium is found on the Mars of the late 19th century, and allows physical activity, unhampered by the thinner martian atmosphere. The Avatar was made aware of its properties by Lt. Dibbs.
However, despite its uses, Oxium is in constant danger of being devoured by the disgusting Oxi-Leeches, for whom Oxium is their favourite food. Even keeping it in a backpack won't save the Oxium from being devoured.
Oxium is also the basic form of currency on Mars. It can be purchased from Calamity Jane at the Trading Post (in either blobs or geodes) in exchange for large amounts of berries, but there is an infinite supply at the Power Plant.