Molecular oxygen is O2. It is two oxygen atoms covalently bonded together by a double bond. The reason that they are not bonded together by a single bond is due to the number of valence electrons in oxygen and also by the octet rule.
The valence electrons for any atom are the electrons in its outermost energy shell. These are the electrons that are available for chemical reactions and bonding. Oxygen has six electrons in its valence shell. This means that two oxygen atoms together have 12 valence electrons. If 4 of the electrons are utilized for a double bond, then that means that 8 electrons are left, or 4 per oxygen atom. So each atom can have 2 lone pairs and the double bond. This sums up to 8 electrons surrounding each atom (4 from the lone pairs and 4 from the double bond), so that gives each oxygen atom an octet of electrons surrounding it, which is the most stable configuration for an atom (a total of 8 electrons fills its valence orbitals).
The octet rule states that all atoms wish to be stable like the Noble Gases and have eight valence electrons.
Oxygen, which is located in the sixth group, only has six valence electrons. That means, it needs an additional two valence electrons to become like a Noble Gas.
Think of the Lewis Dot structure of oxygen. There is one set of lone pairs on the top and another on the bottom. That leaves one electron on the left and one on the right. To create a full octet and oxygen (O2), one of the electrons must be moved to form a double bond (I have linked a picture to provide a visual aid). If there was a single bond instead, each oxygen atom would only have seven valence electrons.