Let's first take a look at the allotropes of oxygen, and look into dioxygen more in depth.
- Atomic oxygen ($\ce{O1}$, a free radical)
- Singlet oxygen ($\ce{O2}$), either of two metastable states of molecular oxygen
- Tetraoxygen ($\ce{O4}$), another metastable form
From NASA, http://www.nasa.gov/topics/technology/features/atomic_oxygen.html, regarding atomic oxygen:
Atomic oxygen doesn't exist naturally for very long on the surface of Earth, as it is very reactive. But in space, where there is plenty of ultraviolet radiation, $\ce{O2}$ molecules are more easily broken apart to create atomic oxygen. The atmosphere in low Earth orbit is comprised of about 96% atomic oxygen. In the early days of NASA's space shuttle missions, the presence of atomic oxygen caused problems.
Dioxygen, or triplet oxygen, is the most commonly known allotrope of oxygen. It has the molecular formula $\ce{O2}$. Oxygen has 8 electrons with 2 in the 1s, 2 in the 2s, 4 in the 3p orbitals. Alternatively, there are 6 valence electrons. If there are another oxygen molecules, oxygen will pair up, form a double bond with bond order of two. In short, the potential energy of dioxygen is far less than that of atomic oxygen.
One interesting aspect of oxygen is that it exhibits paramagnetism unlike $\ce{N2}$ and it can exist in two different electronic states called singlet oxygen. The picture of the molecular orbital (MO) diagram of oxygen makes this more clear:

The MO diagrams above are for the singlet oxygen $a^1\Delta g$ excited state, the singlet oxygen $b^1\Sigma \text{g+}$ excited state, and the triplet ground state $X^3\Sigma \text{g-}$ respectively.
What you may notice is a spin flip in the $b^1\Sigma \text{g+}$ excited state.
This definition taken from Purdue University summarizes Hund's rule of maximum simplicity nicely: every orbital in a subshell is singly occupied with one electron before any one orbital is doubly occupied, and all electrons in singly occupied orbitals have the same spin.
The two first diagrams are in violation of the 1.) spin-selection rule: spin-flips are foridden, and 2.) Laporte selection rule: trasnsitions between orbitals of the same parity are forbidden, where parity means symmetry with respect to inversion. There is a german notation, gerade - which refers to symmetric with respect to inversion and ungerade - antisymmetric with respect to inversion.
There are many ways to produce ozone. https://en.wikipedia.org/wiki/Ozone#Production
Ozone is a triatomic molecule with 3 oxygens. It is much less stable than dioxygen and often breaks down into dioxygen.
What you may have been confused with is Dalton's incorrect "rule of greatest simplicity." Dalton was trying to resolve the issue of the correct ratio and number of atoms in respect to the chemical formula.
He assumed that:
$$ \ce{H + O -> H2O}$$
However, we know that:
$$ \ce{H2 + O2 -> 2H2O}$$
It wasn't until Avogadro and Gay-Lussac who stated the Law of Multiple Proportions and postulated the existence of diatomic molecules, that we can now resolve Dalton's incorrect hypothesis.
When two elements form a series of compounds, the masses of one element that
combine with a fixed mass of the other element are in the ratio of small integers
to each other.