Oxygen is found in 3 forms: nascent (O), molecular (O2, the most common) and ozone (O3). Nascent oxygen, due to its electronic configuration (i.e how many electrons it has and how they're spread out across its electronic shells) is unstable, and tends to quickly form bonds with another O, forming O2. This is also the case e.g. for hydrogen, which is usually found as H2.

You can find O in this form in some environments, in the upper atmosphere there is enough UV radiation to break up O2 into O.

There are 7 elements that will naturally form covalent bonds with themselves.
Here is how to remember these diatomic elements:

(H)ave (N)o (F)ear (O)f (I)ce (Cl)old (Br)eer.
The Ice is solid, the beer is liquid, and everything else is a gas.

Because O is not naturally accuring and only exists under extremely special conditions, so virtually all pure oxygen appears as molecular oxygen O2.

The stuff around us is made out of molecules. Most molecules are made out of more than one atom. Oxygen, the gas, is made out of oxygen molecules. An oxygen molecule is made out of two oxygen atoms. So O2. When you hear "pure oxygen", it's about the substance oxygen (O2), as in pure from other găsesc, not the element oxygen (O).

Now you can also have O or O3 molecules, but those have an electric charge (aka ions), so they're unstable and prone to bond with other elements or to break apart. A gas made out of monoatomic oxygen would be extremely reactive.

Me: mom can we get O?

Mom: we have O at home.

O At home: O2

In its natural gaseous state, oxygen is generally found in pairs, hence O2. I guess when you saw "pure oxygen", this meant gaseous oxygen without any other gas normally found in the atmosphere. So no nitrogen, or carbon dioxide, just oxygen. In the form of O2 of course!

Because O doesn't like to exist. O~2~ is simply more stable. A

Very few elements are composed of individual atoms that arent part of a molecule or crystal in their native form under ambient conditions. Individual Oxygen atoms dont have a full shell of electrons. They are extremely reactive for this reason and will pair up to form the considerably more stable O2 molecule. As for why it is more stable, when atomic orbitals interact with one another to form molecular orbitals, the number of molecular orbitals that form is equal to the number of atomic orbitals used to form them. Generally there are two classes of molecular orbitals: bonding and antibonding. Bonding orbitals contribute constructively to the formation of bonds. Antibonding orbitals contribute destructively to the formation of bonds. I.e they weaken bonds. There are a few analogies that can be used to sort of understand what is going on. Constructive interactions are like pushing someone in a swing at just the right time to add to their back and forth motion OR swinging a jump rope at the right time to maintain the wave. Destructive interactions are essentially the opposite of this. An analogy would be pushing someone in a swing at just the right time to slow their motion until they stop or swinging a jump rope at the right time needed to cause its motion to slow until it stops oscillating. If you have experience with tuning forks, youll also notice that if you couple two vibrating tuning forks together you get two separate sets of possible frequencies. One lower and one higher. This has the effect of stabilizing the system. Something similar happens when molecular orbitals form from the interactions of atomic orbitals from individual atoms forming molecules. You can also visualize this extra stability in the molecular orbital diagram for O2 vs the individual atoms: https://chem.libretexts.org/Courses/Pacific_Union_College/Quantum_Chemistry/09%3A_Chemical_Bonding_in_Diatomic_Molecules/9.10%3A_Molecular_Orbital_Theory_Predicts_that_Molecular_Oxygen_is_Paramagnetic

Except for noble gases, single atoms are usually quite unstable and will prefere to release some energy to settle into a more stable molecular configuration.

O2 denotes two oxygen atoms binding together into one molecule.

O1 or simply O can't really exist for long in nature, because it's not stable and will bind with almost anything it meets, including another O atom.

Oxygen naturally occurs as molecules or two of them joined together to form O~2~

Cause O1 and O3 are not good for breathing.

Dioxygen

I'll tell you why. Big Air is blatantly ripping off the city of London. Why they don't sue for trademark infringement, I don't know.

I prefer to call O2 something else: dioxygen

It's what I learned in chemistry class (same applies to the other diatomic molecular gases).