Singlet oxygen is one of the possible energy states of the oxygen molecule, O2. It is called a singlet state because all of the electrons are spin-paired, so there is only one possible value, 0, for the total spin for the whole molecule. The singlet state has a higher energy than the triplet oxygen state, so it is called an excited state.
The singlet state is the structure of oxygen seen in most introductions to chemistry. It has a double bond between two oxygen atoms, for the structure O=O. This is different from the two three-electron bonds in the triplet state, which can't easily be drawn as a Lewis structure.
Because of conservation of angular momentum, making a chemical bond only happens between two singlet molecules or two triplet molecules. Since most molecules are singlets, this means that triplet oxygen only reacts when it has enough activation energy to change spin (a spin-forbidden reaction). Singlet oxygen does not have this restriction, making it much more reactive.